# AP Calculus AB : Derivatives

## Example Questions

### Example Question #33 : Derivatives Of Functions

Given y(z), find y'(z).

Explanation:

Given y(z), find y'(z).

Now, we have three terms to our function, y(z). To find y'(z), we need to recall three rules.

1)

2)

3)

Using these three rules, we can solve our problem.

1)

2)

3)

Put all our terms together to get:

### Example Question #36 : Derivatives Of Functions

Given y(z), find y''(z).

Explanation:

Given y(z), find y''(z).

Now, we have three terms to our function, y(z). To find y''(z), we need to first find y'(z), then we can differentiate again to get y"(z).

To find y'(z)...

1)

2)

3)

Using these three rules, we can complete the first step.

1)

2)

3)

Put all our terms together to get:

Now, we need to replicate the process to get y"(z).

1) Our first term will remain the same.

2) Our second term will follow the same rule and reduce again.

3) Now we need to recall another rule:

So now we can put it all back together to get:

### Example Question #34 : Derivatives Of Functions

Find the derivative of the following function:

Explanation:

The derivative of the function is equal to

and was found using the following rules:

For a sum, the derivative is simply the sum of the derivatives of the individual parts.

### Example Question #35 : Derivatives Of Functions

Find the first derivative of the following function:

Explanation:

The first derivative of the function is equal to

and was found using the following rules:

### Example Question #36 : Derivatives Of Functions

Find the first derivative of the following function:

Explanation:

The derivative of the function is equal to

and was found using the following rules:

Taking the derivative without simplification gets us

and algebra is used to simplify to the final expression above.

### Example Question #40 : Derivatives Of Functions

Calculate the derivative:

Explanation:

This is a chain rule using trigonometric functions.

Which upon simplifying is:

### Example Question #51 : Ap Calculus Ab

Find the first derivative of the following function:

,

where  are all constants.

Explanation:

The derivative of the function is equal to

and was found using the following rules:

### Example Question #41 : Derivatives Of Functions

Find the second derivative of the following function:

Explanation:

To find the second derivative of the function, we first must find the first derivative, which is equal to

which was found using the following rules:

The second derivative is equal to

and was found using the same rules as above, as well as

### Example Question #53 : Ap Calculus Ab

Find the second derivative of the following function:

Explanation:

First, we find the first derivative:

This was found using the following rules:

Next, find the second derivative:

The following additional rules were used:

### Example Question #52 : Ap Calculus Ab

Find the first derivative of the following function: