# High School Math : Finding Derivatives

## Example Questions

### Example Question #1 : Understanding The Derivative Of Exponents

Find the derivative for

Explanation:

The derivative must be computed using the product rule.  Because the derivative of  brings a  down as a coefficient, it can be combined with  to give

### Example Question #4 : Understanding The Derivative Of Exponents

Give the instantaneous rate of change of the function  at .

Explanation:

The instantaneous rate of change of  at  is , so we will find  and evaluate it at .

for any positive , so

### Example Question #2 : Understanding The Derivative Of Exponents

What is  ?

Explanation:

Therefore,

for any real , so , and

### Example Question #3 : Understanding The Derivative Of Exponents

What is  ?

Explanation:

Therefore,

for any positive , so , and

### Example Question #1 : Understanding The Derivative Of Trigonometric Functions

Find the derivative of the following function:

Explanation:

The derivative of  is. It is probably best to memorize this fact (the proof follows from the difference quotient definition of a derivative).

Our function

the factor of 3 does not change when we differentiate, therefore the answer is

### Example Question #2 : Understanding The Derivative Of Trigonometric Functions

Explanation:

The derivative of a sine function does not follow the power rule. It is one that should be memorized.

.

### Example Question #3 : Understanding The Derivative Of Trigonometric Functions

What is the second derivative of ?

Explanation:

The derivatives of trig functions must be memorized. The first derivative is:

.

To find the second derivative, we take the derivative of our result.

.

Therefore, the second derivative will be .

### Example Question #5 : Understanding The Derivative Of Trigonometric Functions

Compute the derivative of the function .

Explanation:

Use the Chain Rule.

Set  and substitute.

### Example Question #1 : Understanding The Derivative Of A Sum, Product, Or Quotient

Find the derivative of the following function:

Explanation:

Since this function is a polynomial, we take the derivative of each term separately.

From the power rule, the derivative of

is simply

We can rewrite  as

and using the power rule again, we get a derivative of

or

What is