# Precalculus : Derivatives

## Example Questions

### Example Question #1 : Find The First Derivative Of A Function

We consider the function

What is the first derivative of ?

Explanation:

Note that  is defined for all

Using Logarithm Laws we can write as .

Using the Chain Rule, and the Product Rule . We have:

.

Note that,

.

Now replace  and we get our final answer:

### Example Question #2 : Find The First Derivative Of A Function

Find the derivative of .

Explanation:

This uses the simple Exponential Rule of derivatives.

Mutiply by the value of the exponent to the function, then subtract 1 from the old exponent to make the new exponent.

The formula is as follows:

.

Using our function,

where  and  we get the following derivative,

.

### Example Question #3 : Find The First Derivative Of A Function

Find the derivative of .

Explanation:

For this problem we need to use the Chain Rule.

The Chain Rule states to work from the outside in. In this case the outside function is  and the inside function is . The derivative then becomes the outside function times the derivative of the inside function.

Thus, we use the following formula

.

In our case , and .

Therefore the result is,

.

### Example Question #4 : Find The First Derivative Of A Function

Find the first derivative of the following function:

Explanation:

In order to take the first derivative of the polynomial, all we need to know is how to apply the power rule to a simple term with an exponent:

The formula above tells us that to take the derivative of a term with coefficient  and exponent , we simply multiply the term by and subtract 1 from in the exponent. With this in mind, we'll take the derivative of the function in the problem term by term:

### Example Question #5 : Find The First Derivative Of A Function

Find the derivative of the following function:

Explanation:

We can see that our function involves a series of terms raised to a power, so we will need to apply the power rule as well as the chain rule to find the derivative of the function. First we apply the power rule to the terms in parentheses as a whole, and then we apply the chain rule by multiplying that entire result by the derivative of the expression in parentheses alone:

### Example Question #6 : Find The First Derivative Of A Function

What is the first derivative of

with regards to ?

Explanation:

Since we take the derivative with respect to  we only apply power rule to terms that contain . Since all the terms contain , we treat  as a constant.

The derivative of a constant is always .

So the derivative of

is zero.

### Example Question #7 : Find The First Derivative Of A Function

Find the first derivative of

with respect to

Explanation:

The derivative of a constant is .

### Example Question #8 : Find The First Derivative Of A Function

Find the first derivative of

with respect to

Explanation:

Recall the power rule that states to multiply by the exponent in front of the constant and then subtract the exponent by 1.

So lets take the derivative of this in sections:

Derivative of :

Result is:

Lets take the derivative of the next term:

Derivative of :

The power of something to  is .

Our result is:

### Example Question #9 : Find The First Derivative Of A Function

Find the first derivative of the function

.

Explanation:

For the function

The first derivative is

So for

the first derivative is

### Example Question #10 : Find The First Derivative Of A Function

Find the first derivative of

in relation to .

Explanation:

To find the derviative of this equation recall the power rule that states: Multiply the exponent in front of the constant and then subtract one from the exponent.

We can work individually with each term:

Derivative of

is,

For the next term:

Derivative of :

Anything to a power of 0 is 1.

For the next term:

Derivative of :

Any derivative of a constant is .

So the first derivative of

is