# AP Calculus BC : Chain Rule and Implicit Differentiation

## Example Questions

### Example Question #41 : Computation Of Derivatives

Explanation:

According to the chain rule, . In this case,  and . The derivative is .

### Example Question #11 : Chain Rule And Implicit Differentiation

Explanation:

According to the chain rule, . In this case,  and . The derivative is

### Example Question #12 : Chain Rule And Implicit Differentiation

Explanation:

According to the chain rule, . In this case,  and  and

The derivative is

### Example Question #13 : Chain Rule And Implicit Differentiation

Explanation:

According to the chain rule, . In this case,  and . Since  and , the derivative is

### Example Question #14 : Chain Rule And Implicit Differentiation

Explanation:

According to the chain rule, . In this case,  and . Since  and , the derivative is

### Example Question #15 : Chain Rule And Implicit Differentiation

Explanation:

According to the chain rule, . In this case,  and . Here  and . The derivative is:

### Example Question #16 : Chain Rule And Implicit Differentiation

Given the relation , find .

Explanation:

We begin by taking the derivative of both sides of the equation.

.

. (The left hand side uses the Chain Rule.)

.

.

### Example Question #17 : Chain Rule And Implicit Differentiation

Given the relation , find .

Explanation:

We can use implicit differentiation to find . We being by taking the derivative of both sides of the equation.

(This line uses the product rule for the derivative of .)

### Example Question #18 : Chain Rule And Implicit Differentiation

If , find .

Explanation:

Since we have a function inside of a another function, the chain rule is appropriate here.

The chain rule formula is

.

In our function, both  are

So we have

and

.

### Example Question #19 : Chain Rule And Implicit Differentiation

Find  of the following:

Explanation:

To find  we must use implicit differentiation, which is an application of the chain rule.
Taking  of both sides of the equation, we get

using the following rules:

Note that for every derivative with respect to x of a function with y, the additional term dy/dx appears; this is because of the chain rule, where y=g(x), so to speak, for the function it appears in.

Using algebra, we get