# AP Calculus AB : Chain rule and implicit differentiation

## Example Questions

### Example Question #41 : Chain Rule And Implicit Differentiation

Find the derivative of the following equation:

Explanation:

Because we are differentiating a function within another function, we must use the chain rule, which states that

.

Given the equation

,

we can deduce that

and

.

By plugging these into the chain rule, we conclude that

.

### Example Question #42 : Chain Rule And Implicit Differentiation

Find the derivative of the function

Explanation:

We proceed as follows.

. (Start)

. (Product rule)

. (The first derivative uses the Chain rule. The 2nd one uses the basic power rule.)

.

.

Although some factoring could be done at this point, we will not do so.

### Example Question #43 : Chain Rule And Implicit Differentiation

Find the derivative of the following function:

.

Explanation:

For a chain rule derivative, we need to work our way inward from the very outermost function.  First, we need to do a power rule for the outer exponent.  Then, we multiply that by the derivative of the inside.

### Example Question #44 : Chain Rule And Implicit Differentiation

Find the derivative of the following function:

.

Explanation:

For a chain rule derivative, we take the derivative of the outside function (leaving the inside function unchanged).  Then, you multiply that by the derivative of the inside.

### Example Question #45 : Chain Rule And Implicit Differentiation

Find the derivative of the following function:

.

Explanation:

For a chain rule derivative, we need to take the derivative of  first.  Then, we multiply by the derivative of the inside function.  In other words, the general chain derivative of the function is:

### Example Question #46 : Chain Rule And Implicit Differentiation

Find the derivative of the following function:

.

Explanation:

The exponential function is the only derivative that always returns the original function.  Therefore, we only need to multiply that by the derivative of the new exponent.

.

### Example Question #47 : Chain Rule And Implicit Differentiation

Find the derivative of the following function:

.

Explanation:

Here, we need to take the derivative of a square root function first.  Then, we will multiply that by the derivative of the function under the square root.

Remember:  by another chain rule.

Now, let's make use of our trigonometric identities:

Therefore, we can simplify our answer:

Note, you could have used this trigonometric identity from the very beginning, making the problem much easier!

### Example Question #48 : Chain Rule And Implicit Differentiation

Find

Explanation:

For implicit differentiation, we need to take a derivative from left to right of our function.  The only difference is that any time we take the derivative of a  function, we need to explicitly write  after it.  Then, we will use algebra to solve for that

Here, just move everything other than the term we want to the right. Then, divide by the coefficient.

### Example Question #49 : Chain Rule And Implicit Differentiation

Find  for the following function:

Explanation:

For implicit differentiation, we need to take the derivative of every term, including numbers.  Whenever we take a derivative of a function of , we need to multiply it by .  Then, we will rearrange the equation to solve for

In the last step, I canceled all of the negative signs.

### Example Question #50 : Chain Rule And Implicit Differentiation

Determine the derivative of

Explanation:

This is a pure problem on understanding how chain rules work for derivatives.

First thing we need to remember is that the derivative of is .

When we are taking the derivative of , we can first pull out the 2 in the front and we treat  as .

This way, the derivative will become ,

which is .