AP Calculus AB : Chain rule and implicit differentiation

Example Questions

Example Question #81 : Chain Rule And Implicit Differentiation

Find the derivative of the function:

Explanation:

The derivative of the function is equal to

and was found using the following rules:

Note that the chain rule was used on the natural logarithm derivative and the derivative of the cosine function.

Example Question #82 : Chain Rule And Implicit Differentiation

Compute the second derivative of the following function:

Explanation:

The first step in finding our second derivative is finding the first.

For our function  we note that it is a composite function, and therefore requires the use of the chain rule. The chain rule states if  then .

In our case this becomes:

To find the second derivative we must take the derivative of this function we've now computed. To do this we will require both the chain rule as stated, and the product rule since our function is of the form , our derivative will be of the form .

Therefore, our second derivative is:

or equivalently,

Example Question #83 : Chain Rule And Implicit Differentiation

Given the relation , find .

Explanation:

We start by taking the derivative of both sides of the equation, and proceeding as follows,

.

Example Question #84 : Chain Rule And Implicit Differentiation

and

Given that

find the derivative of .

Explanation:

First step is to figure out what  is. This is done by inputting  wherever there is an  in , which gives you

.

Then you start on the outermost function, the sine function, and work your way inwards.

Again, chain rule for  is:

,

which gives us for this problem

Example Question #85 : Chain Rule And Implicit Differentiation

Find  for the equation.

Explanation:

So implicit differentiation means that you take the derivative of every variable in the same way, with its respective rate of change, dx or dy in this case, coming out multiplied on the end.

You take the derivative of the whole equation and the result is:

The first step to solve for dy/dx is to create a dy/dx term, which means that you need to divide the entire equation by dx, to get it on the bottom. Each term becomes divided by dx, so the dx cancels with itself and you are left with two terms that have . The result is:

At this point, you want to get  on its own side, so this involves subtracting the two terms that aren't multiplied by  onto the right side of the equation.

After that, you factor out the  and end up with:

Then, the last step is to divide over the two term block to the other side to get  on its own.

Example Question #85 : Chain Rule And Implicit Differentiation

and

Find the derivative of

Explanation:

The chain rule of the derivative always deals with the composition of two or more functions.

In this case we can identify two,

and

.

So  is the composition of these two such that:

With chain rule, you always start on the outermost function and work your way inward, which in this case is:

Always the derivative of the outermost evaluated at the inner, multiplied by the derivative of the inner.

Example Question #86 : Chain Rule And Implicit Differentiation

Find

Explanation:

We are going to use three rules along with the chain rule:

So then, using our first rule and the chain rule

then using our second rule and chain rule

then using our third rule (no chain rule this time)

Then we rearrange the equation for simplification,

Example Question #87 : Chain Rule And Implicit Differentiation

Find

Explanation:

We are going to use two rule and the chain rule

Then, using rule one and rule two (don't forget the chain rule)

Then we simplify

Example Question #88 : Chain Rule And Implicit Differentiation

Find

Explanation:

We will use the chain rule combined with our power rule:

Then by using this rule

Then applying the power rule to each element in the second parenthesis,

Example Question #90 : Chain Rule And Implicit Differentiation

Find

Explanation:

We will be using the chain rule along with the following rules:

Looking at , first we will apply rule 3 with the chain rule,

Now we apply rule 1 (with chain rule) and rule 3 (no chain rule needed here),

Lastly we apply rule 2 (no chain rule needed),

Then we simplify as much as we can, and of course, what we have just calculated is the derivative of the original,