AP Calculus AB : Equations involving derivatives

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

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Example Question #1 : Equations Involving Derivatives

Practicing the chain rule level 1 B!

Find the derivative of the function

Possible Answers:

Correct answer:

Explanation:

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

Chain rule table

Example Question #2 : Equations Involving Derivatives

Practicing the chain rule level 1 A!

Find the derivative of the function

Possible Answers:

Correct answer:

Explanation:

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

Chain rule table

 

 

Example Question #3 : Equations Involving Derivatives

Practicing the chain rule level 1 C!

Find the derivative of the function

Possible Answers:

Correct answer:

Explanation:

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

Chain rule table

Example Question #4 : Equations Involving Derivatives

Practicing the chain rule level 1 D!

Find the derivative of the function

Possible Answers:

Correct answer:

Explanation:

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

Chain rule table

Example Question #5 : Equations Involving Derivatives

Practicing the chain rule level 2 A!

Find the derivative of the function

Possible Answers:

Correct answer:

Explanation:

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

Chain rule table

Example Question #6 : Equations Involving Derivatives

Practicing the chain rule level 2 B!

Find the derivative of the function

Possible Answers:

Correct answer:

Explanation:

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

Chain rule table

Example Question #7 : Equations Involving Derivatives

Practicing the chain rule level 2 C!

Find the derivative of the function

Possible Answers:

Correct answer:

Explanation:

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

Chain rule table

Example Question #8 : Equations Involving Derivatives

Practicing the chain rule level 2 D!

Find the derivative of the function

Possible Answers:

Correct answer:

Explanation:

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

Chain rule table

Example Question #9 : Equations Involving Derivatives

Practicing the chain rule level 3 A!

Find the derivative of the function

Possible Answers:

Correct answer:

Explanation:

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.

Chain rule table

Example Question #10 : Equations Involving Derivatives

Practicing the chain rule level 3 B!

Find the derivative of the function

Possible Answers:

Correct answer:

Explanation:

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.

Chain rule table

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