# AP Calculus AB : Chain rule and implicit differentiation

## Example Questions

### Example Question #21 : Chain Rule And Implicit Differentiation

Differentiate

Explanation:

In this chain rule derivative, we need to remember what the derivative of the base function ().  The exponential function is a special function that always returns the original function.  In the chain rule application, we just need to multiply that by the derivative of the exponent.

.

### Example Question #22 : Chain Rule And Implicit Differentiation

Find the derivative of .

Explanation:

Be careful with this derivative because it is a hidden chain rule! Let's start with rewriting the problem:

Now, it is easier to notice that it is a chain rule.  A chain rule involves differentiating the outermost function and then, multiplying by the derivative of the inner part.

### Example Question #23 : Chain Rule And Implicit Differentiation

Compute the derivative of

Explanation:

This is a multiple chain derivative.  With chain derivatives, we always want to start on the outermost function (keeping the rest the same), and then, multiply by the inner function derivative until we have take the derivative of every part.

### Example Question #201 : Derivatives

Find the derivative:

Explanation:

Implicit differentiation is useful when we have an equation containing a dependent variable , and and an independent variable , but we cannot solve to write  explicitly in terms of

Differentiate both sides simultaneously:

The chain rule must be applied to the following derivatives:

Noting the  is implied to be a function of , we have to first differentiate with respect to  and then differentiate  with respect to  Since we do not have an explicit definition for , we simply express its' derivative in the equation as , treating it as an "unknown," which needs to be solved for algebraically at a later step.

For (1):

For (2):

Using these derivatives the main equation becomes:

Now we simply solve for  and simplify as much as possible. Isolate all terms with a derivative onto one side of the equation and factor:

### Example Question #24 : Chain Rule And Implicit Differentiation

Find the derivative of the function:

Explanation:

On this problem we have to use chain rule, which is:

So in this problem we let

and

.

Since

and

,

we can conclude that

### Example Question #25 : Chain Rule And Implicit Differentiation

Find the derivative of:

Explanation:

On this problem we have to use chain rule, which is:

So in this problem we let

and

.

Since

and

,

we can conclude that

### Example Question #26 : Chain Rule And Implicit Differentiation

Find the derivative of:

Explanation:

On this problem we have to use chain rule, which is:

So in this problem we let

and

.

Since

and

,

we can conclude that

and

### Example Question #27 : Chain Rule And Implicit Differentiation

Find the derivative of:

Explanation:

On this problem we have to use chain rule, which is:

So in this problem we let

and

and

.

Since

and

and

,

we can conclude that

### Example Question #28 : Chain Rule And Implicit Differentiation

Find the derivative of the function:

Explanation:

On this problem we have to use chain rule, which is:

So in this problem we let

and

.

Since

and

,

we can conclude that

and

### Example Question #29 : Chain Rule And Implicit Differentiation

Find the derivative of:

Explanation:

On this problem we have to use chain rule, which is:

So in this problem we let

and

.

Since

and

,

we can conclude that