Equations involving derivatives - AP Calculus AB
Card 1 of 143
Find the eqation of the line tangent to the graph of $y=x+x^{1/2}$ at point 
.
Find the eqation of the line tangent to the graph of $y=x+x^{1/2}$ at point
Tap to reveal answer
$y=x+x^{1/2}$
$y'=1+\frac{1}{2}$x^{-1/2}$
$y'=1+\frac{1}{2(x^{1/2}$$)}
Now we plug in 
.
$y'=1+\frac{1}{2(4^{1/2}$$)} = 1+\frac{1}{4}$ = $\frac{5}{4}$ this is the slope.
Now use the point slope formula to find the tangent line.
y-6=\frac{5}{4}$(x-4)
y-6=\frac{5}{4}$x-5
y=\frac{5}{4}$x+1
$y=x+x^{1/2}$
$y'=1+\frac{1}{2}$x^{-1/2}$
$y'=1+\frac{1}{2(x^{1/2}$$)}
Now we plug in
$y'=1+\frac{1}{2(4^{1/2}$$)} = 1+\frac{1}{4}$ = $\frac{5}{4}$ this is the slope.
Now use the point slope formula to find the tangent line.
y-6=\frac{5}{4}$(x-4)
y-6=\frac{5}{4}$x-5
y=\frac{5}{4}$x+1
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Practicing the chain rule level 1 A!
Find the derivative of the function
Practicing the chain rule level 1 A!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and 
.
in terms of 
and 
is actually 
, which means
since 
in 
is substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
← Didn't Know|Knew It →
Practicing the chain rule level 1 B!
Find the derivative of the function
Practicing the chain rule level 1 B!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and 
.
in terms of and is actually , which means
since in 
is substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the component composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the component composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
← Didn't Know|Knew It →
Practicing the chain rule level 1 C!
Find the derivative of the function
Practicing the chain rule level 1 C!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and 
.
in terms of and is actually , which means
since in 
is substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the component composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the component composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
← Didn't Know|Knew It →
Practicing the chain rule level 1 D!
Find the derivative of the function
Practicing the chain rule level 1 D!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and 
.
in terms of 
and 
is actually 
and
since in 
is substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the component composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the function g(x) but multiply its derivative by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the component composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the function g(x) but multiply its derivative by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
← Didn't Know|Knew It →
Practicing the chain rule level 1 E!
Find the derivative of the function
Practicing the chain rule level 1 E!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
And finally, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
, 
and 
.
The reason why we did not define 
, 
and 
is because of the property 
mentioned before,
turning
into
in terms of , and 
is actually 
which means
since in 
is substituted with 
and 
in 
can be substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Look at the next function h(x), keep it inside the other function g(x)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(x) but multiply it by the factors f'(g(h(x)))*g'(h(x))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(x), g(x), g'(x), h(x) and h'(x) for the expressions you found before:
which is 
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
And finally, you must understand that the derivative of
is actually
The reason why we did not define
is because of the property
turning
into
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Look at the next function h(x), keep it inside the other function g(x)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(x) but multiply it by the factors f'(g(h(x)))*g'(h(x))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(x), g(x), g'(x), h(x) and h'(x) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
← Didn't Know|Knew It →
Practicing the chain rule level 2 A!
Find the derivative of the function
Practicing the chain rule level 2 A!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and 
.
in terms of and is actually and
since in 
is substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the function g(x) but multiply its derivative by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the function g(x) but multiply its derivative by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
← Didn't Know|Knew It →
Practicing the chain rule level 2 B!
Find the derivative of the function
Practicing the chain rule level 2 B!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and 
.
in terms of 
and 
is actually 
which means
since 
in 
is substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(t), g(t) and g'(t) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(t), g(t) and g'(t) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
← Didn't Know|Knew It →
Practicing the chain rule level 2 C!
Find the derivative of the function
Practicing the chain rule level 2 C!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and 
.
in terms of and is actually which means
since in 
is substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
← Didn't Know|Knew It →
Practicing the chain rule level 2 D!
Find the derivative of the function
Practicing the chain rule level 2 D!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and 
.
in terms of and is actually which means
since in 
is substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
← Didn't Know|Knew It →
Practicing the chain rule level 3 A!
Find the derivative of the function
Practicing the chain rule level 3 A!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
first remember that 
.
Then, you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
And finally, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
, and .
The reason why we did not define , and
is because of the property mentioned before,
turning
into
in terms of , and is actually which means
since in is substituted with and in can be substituted with .
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Look at the next function h(x), keep it inside the other function g(x)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(x) but multiply it by the factors f'(g(h(x)))*g'(h(x))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(x), g(x), g'(x), h(x) and h'(x) for the expressions you found before:
which is
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
To understand why the answer is
first remember that
Then, you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
And finally, you must understand that the derivative of
is actually
The reason why we did not define
is because of the property
turning
into
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Look at the next function h(x), keep it inside the other function g(x)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(x) but multiply it by the factors f'(g(h(x)))*g'(h(x))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(x), g(x), g'(x), h(x) and h'(x) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
← Didn't Know|Knew It →
Practicing the chain rule level 3 B!
Find the derivative of the function
Practicing the chain rule level 3 B!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
And finally, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
, 
and 
.
in terms of 
, 
and 
is actually 
which means
since 
in 
is substituted with 
and 
in 
can be substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(z) first, then differentiate it
Step 2: Look at the next function g(z), keep it inside the other function f'(z)
Step 3: Look at the next function h(z), keep it inside the other function g(z)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(z) but multiply it by the factors f'(g(h(z)))*g'(h(z))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(z), g(z), g'(z), h(z) and h'(z) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to differentiate. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
And finally, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(z) first, then differentiate it
Step 2: Look at the next function g(z), keep it inside the other function f'(z)
Step 3: Look at the next function h(z), keep it inside the other function g(z)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(z) but multiply it by the factors f'(g(h(z)))*g'(h(z))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(z), g(z), g'(z), h(z) and h'(z) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to differentiate. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
← Didn't Know|Knew It →
Practicing the chain rule level 3 C!
Find the derivative of the function
Practicing the chain rule level 3 C!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
Then, you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
And finally, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
, 
and 
.
in terms of , and is actually which means
since in 
is substituted with 
and in 
can be substituted with 
.
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Look at the next function h(x), keep it inside the other function g(x)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(x) but multiply it by the factors f'(g(h(x)))*g'(h(x))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(x), g(x), g'(x), h(x) and h'(x) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
To understand why the answer is
Then, you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
And finally, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Look at the next function h(x), keep it inside the other function g(x)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(x) but multiply it by the factors f'(g(h(x)))*g'(h(x))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(x), g(x), g'(x), h(x) and h'(x) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
← Didn't Know|Knew It →
Practicing the chain rule level 3 A!
Find the derivative of the function
Practicing the chain rule level 3 A!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
first remember that .
Then, you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
And finally, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
, and .
The reason why we did not define , and
is because of the property mentioned before,
turning
into
in terms of , and is actually which means
since in is substituted with and in can be substituted with .
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Look at the next function h(x), keep it inside the other function g(x)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(x) but multiply it by the factors f'(g(h(x)))*g'(h(x))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(x), g(x), g'(x), h(x) and h'(x) for the expressions you found before:
which is
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
To understand why the answer is
first remember that
Then, you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
And finally, you must understand that the derivative of
is actually
The reason why we did not define
is because of the property
turning
into
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Look at the next function h(x), keep it inside the other function g(x)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(x) but multiply it by the factors f'(g(h(x)))*g'(h(x))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(x), g(x), g'(x), h(x) and h'(x) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
← Didn't Know|Knew It →
Practicing the chain rule level 3 B!
Find the derivative of the function
Practicing the chain rule level 3 B!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
And finally, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
, and .
in terms of , and is actually which means
since in is substituted with and in can be substituted with .
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(z) first, then differentiate it
Step 2: Look at the next function g(z), keep it inside the other function f'(z)
Step 3: Look at the next function h(z), keep it inside the other function g(z)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(z) but multiply it by the factors f'(g(h(z)))*g'(h(z))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(z), g(z), g'(z), h(z) and h'(z) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to differentiate. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
And finally, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(z) first, then differentiate it
Step 2: Look at the next function g(z), keep it inside the other function f'(z)
Step 3: Look at the next function h(z), keep it inside the other function g(z)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(z) but multiply it by the factors f'(g(h(z)))*g'(h(z))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(z), g(z), g'(z), h(z) and h'(z) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to differentiate. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
← Didn't Know|Knew It →
Find the eqation of the line tangent to the graph of $y=x+x^{1/2}$ at point .
Find the eqation of the line tangent to the graph of $y=x+x^{1/2}$ at point
Tap to reveal answer
$y=x+x^{1/2}$
$y'=1+\frac{1}{2}$x^{-1/2}$
$y'=1+\frac{1}{2(x^{1/2}$$)}
Now we plug in .
$y'=1+\frac{1}{2(4^{1/2}$$)} = 1+\frac{1}{4}$ = $\frac{5}{4}$ this is the slope.
Now use the point slope formula to find the tangent line.
y-6=\frac{5}{4}$(x-4)
y-6=\frac{5}{4}$x-5
y=\frac{5}{4}$x+1
$y=x+x^{1/2}$
$y'=1+\frac{1}{2}$x^{-1/2}$
$y'=1+\frac{1}{2(x^{1/2}$$)}
Now we plug in
$y'=1+\frac{1}{2(4^{1/2}$$)} = 1+\frac{1}{4}$ = $\frac{5}{4}$ this is the slope.
Now use the point slope formula to find the tangent line.
y-6=\frac{5}{4}$(x-4)
y-6=\frac{5}{4}$x-5
y=\frac{5}{4}$x+1
← Didn't Know|Knew It →
Practicing the chain rule level 3 C!
Find the derivative of the function
Practicing the chain rule level 3 C!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
Then, you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
And finally, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
, and .
in terms of , and is actually which means
since in is substituted with and in can be substituted with .
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Look at the next function h(x), keep it inside the other function g(x)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(x) but multiply it by the factors f'(g(h(x)))*g'(h(x))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(x), g(x), g'(x), h(x) and h'(x) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
To understand why the answer is
Then, you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
And finally, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable one step at a time and multiply the results together.
the derivative of
is...
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Look at the next function h(x), keep it inside the other function g(x)
Step 4: Differentiate the the next function g(x) while keeping h(x) inside of g(x) and g'(x). But, multiply g'(h(x)) by f'(g(h(x)))
Step 5: Differentiate the next function h(x) but multiply it by the factors f'(g(h(x)))*g'(h(x))
Since you are out of composite functions to differentiate, stop here.
Now, substitute f'(x), g(x), g'(x), h(x) and h'(x) for the expressions you found before:
and now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner functions while keeping the next inner functions the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in the table below.
← Didn't Know|Knew It →
Practicing the chain rule level 1 A!
Find the derivative of the function
Practicing the chain rule level 1 A!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and .
in terms of and is actually , which means
since in is substituted with .
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
← Didn't Know|Knew It →
Practicing the chain rule level 1 B!
Find the derivative of the function
Practicing the chain rule level 1 B!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and .
in terms of and is actually , which means
since in is substituted with .
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the component composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the component composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
← Didn't Know|Knew It →
Practicing the chain rule level 1 C!
Find the derivative of the function
Practicing the chain rule level 1 C!
Find the derivative of the function
Tap to reveal answer
To understand why the answer is
,
you must understand that the derivative of
is actually
.
Next, you must understand that the derivative of
is actually
.
can be treated as a composition of the functions
and .
in terms of and is actually , which means
since in is substituted with .
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the component composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
-------------------------------------------------------------------------------------------
To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
To understand why the answer is
you must understand that the derivative of
is actually
Next, you must understand that the derivative of
is actually
This means in order to differentiate the equation, you must use the chain rule. The chain rule tells us that in order to differentiate a composition function, you must differentiate all of the component composite functions in the reverse order that they are applied to the variable, one step at a time and multiply the results together.
the derivative of
is
Step 1: Only look at the outermost function f(x) first, then differentiate it
Step 2: Look at the next function g(x), keep it inside the other function f'(x).
Step 3: Differentiate the the next function g(x) but multiply it by f'(g(x))
So, substitute f'(x), g(x) and g'(x) for the expressions you found before:
And now you have found the correct answer.
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To sum it up, you multiply the derivative of the outermost functions by the derivative if the inner function while keeping the next inner functions the the same in every step until you are out of functions to find the derivative of. This also applies to more complicated functions. For instance,
If we have
then it's derivative would be
Notice how the factors become less complicated as you differentiate it or as you look from left to right.
If you are unsure of the pattern, look at the pattern in these equations
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