# AP Calculus AB : Derivatives

## Example Questions

### Example Question #55 : Derivatives Of Functions

Use the method of your choice to find the derivative.

Explanation:

The easiest way to find this derivative is to FOIL, and then use the power rule.

### Example Question #56 : Derivatives Of Functions

Find the derivative.

Explanation:

Use the product rule to find this derivative.

### Example Question #57 : Derivatives Of Functions

Define

Evaluate  and  so that  is both continuous and differentiable at .

Explanation:

For  to be continuous at , it must hold that

.

To find , we can use the definition of  for all negative values of :

It must hold that  as well; using the definition of  for all positive values of :

.

Therefore, .

Now examine . For  to be differentiable, it must hold that

To find , we can differentiate the expression for  for all negative values of :

Again, through straightforward substitution,

To find , we can differentiate the expression for  for all positive values of :

Again, through substitution,

and .

### Example Question #58 : Derivatives Of Functions

Find the derivative.

Explanation:

Use the power rule to find the derivative.

Thus, the derivative is

### Example Question #59 : Derivatives Of Functions

Find the derivative.

Explanation:

Use the quotient rule to find the derivative.

Simplify.

### Example Question #71 : Ap Calculus Ab

Find the first derivative of the function:

Explanation:

The derivative of the function is equal to

and was found using the following rules:

### Example Question #71 : Ap Calculus Ab

Given . Find .

Explanation:

First, find the first derivative.

You should get .

Next, differentiate again.

You should get .

Finally, plug in x=2 to get .

### Example Question #72 : Ap Calculus Ab

The velocity profile of a fluid is given by

Determine the rate of change of the velocity of the fluid at any point.

Explanation:

To find the rate of change of the velocity of the fluid at any point, we must take the derivative of the function:

The derivative was found using the following rules:

### Example Question #73 : Ap Calculus Ab

Find :

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 square root, the inner tangent function, and the function inside the secant.

### Example Question #71 : Ap Calculus Ab

Find the derivative of the function:

Explanation:

The derivative of the function is equal to

and was found using the following rules:

Notice that the chain rule was used on the cosine squared function, the cosine function, and the function inside the sine function.