# Precalculus : Derivatives

## Example Questions

### Example Question #31 : Find The First Derivative Of A Function

Find the derivative:

Explanation:

The derivative of this function, known as  (read as f prime of x), can be found by using the Power rule of derivatives on each term in the function. Power rule: . If there is a value in front of x we multiply it by the "n" we carried over. For example taking the derivative of .

Using the Power rule:

Note that the derivative of any constant is simply 0.

### Example Question #31 : Find The First Derivative Of A Function

Find the first derivative of the function

Explanation:

Use the product rule to find the first derivative.

Here, let  and

### Example Question #33 : Find The First Derivative Of A Function

Find the first derivative of the function

.

Explanation:

Using the power rule,

Apply the Chain Rule.

### Example Question #34 : Find The First Derivative Of A Function

and . Find the derivative of .

Explanation:

Since we have the product of two different expressions we will use the product rule:

Using the product rule:

Combine and simplify:

### Example Question #35 : Find The First Derivative Of A Function

What is the derivative of the following function:

with respect to

Explanation:

This question is meant to test your understanding of derivatives.

Since the term  is completely unrelated to the variable

### Example Question #31 : Find The First Derivative Of A Function

Determine the first derivative of .

Explanation:

Use the power rule to solve this derivative.

### Example Question #37 : Find The First Derivative Of A Function

Calculate the derivative of .

Explanation:

The derivative of constants, or numbers, are zero.  The term  has no variables and will simplify into some number if the  was replaced with .

The result of  will be a constant.  Therefore, the derivative, or slope, of  is zero.

### Example Question #38 : Find The First Derivative Of A Function

Find the derivative of:

Explanation:

To find the derivative of this function, use the power rule and the chain rule.

The power rule is:

The chain rule is to take the derivative of the inner function.

Apply the power rule for the function and the chain rule.  The derivative of  is .

### Example Question #39 : Find The First Derivative Of A Function

Find the first derivative of .

Explanation:

To simplify, simply take the derivative according to the rules for derivatives. Thus,

### Example Question #40 : Find The First Derivative Of A Function

Find the first derivative of the following function:

Explanation:

To solve, simply use the power rule and rule for differentiating natural log as outlined below.

Power rule:

Differentiating natural log:

Thus,

Thus, our first derivative is: