# AP Calculus AB : Chain rule and implicit differentiation

## Example Questions

### Example Question #241 : Derivatives

Given the function , find its derivative.

Explanation:

Given the function , we can find its derivative using the chain rule, which states that

where  and   for . We have  and , which gives us

### Example Question #242 : Derivatives

Given the function , find its derivative.

Explanation:

Given the function , we can find its derivative using the chain rule, which states that

where  and   for . We have  and , which gives us

### Example Question #243 : Derivatives

Given the function , find its derivative.

Explanation:

Given the function , we can find its derivative using the chain rule, which states that

where  and   for . We have  and , which gives us

### Example Question #61 : Chain Rule And Implicit Differentiation

Given the function , find its derivative.

Explanation:

Given the function , we can find its derivative using the chain rule, which states that

where  and   for . We have  and , which gives us

### Example Question #65 : Chain Rule And Implicit Differentiation

Find .

Explanation:

Let

Then

can be rewritten as

Let

The function can now be rewritten as

Applying the chain rule twice:

### Example Question #65 : Chain Rule And Implicit Differentiation

Find the derivative of the function

Explanation:

To find the derivative of the function, you must apply the chain rule, which is as follows:

Using the function from the problem statement, we have that

and

Following the rule, we get

### Example Question #66 : Chain Rule And Implicit Differentiation

Find the derivative of the function

Explanation:

To find the derivative of the function, you must apply the chain rule, which is as follows:

Using the function from the problem statement, we have that

and

Following the rule, we get

### Example Question #161 : Computation Of The Derivative

Find .

None of the other choices gives the correct response.

Explanation:

Let

Then

and

Apply the chain rule:

Substitute back for :

Apply the sum rule:

After some simple algebra:

### Example Question #69 : Chain Rule And Implicit Differentiation

is a function of . Solve for  in this differential equation:

Explanation:

The expressions with  can be separated from those with  by multiplying both sides by :

Find the indefinite integral of both sides:

Set . Then , or , and

Substitute back:

;

Raise  to both powers:

.

The correct choice is

### Example Question #70 : Chain Rule And Implicit Differentiation

Find the derivative using the chain rule.