# High School Math : Specific Derivatives

## Example Questions

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### Example Question #1 : Using Implicit Differentiation

An ellipse is represented by the following equation:

What is the slope of the curve at the point (3,2)?

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Explanation:

It would be difficult to differentiate this equation by isolating . Luckily, we don't have to.  Use to represent the derivative of  with respect to and follow the chain rule.

(Remember, is the derivative of  with respect to , although it usually doesn't get written out because it is equal to 1. We'll write it out this time so you can see how implicit differentiation works.)

Now we need to isolate by first putting all of these terms on the same side:

This is the equation for the derivative at any point on the curve. By substituting in (3, 2) from the original question, we can find the slope at that particular point:

### 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 #1 : Understanding Derivatives 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 #1 : Understanding The Derivative Of Exponents

What is  ?

Explanation:

Therefore,

for any real , so , and

### Example Question #2 : 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 Derivatives 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 : Specific Derivatives

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 #4 : Specific Derivatives

Compute the derivative of the function .

Explanation:

Use the Chain Rule.

Set  and substitute.

### Example Question #1 : Understanding Derivatives Of Sums, Quotients, And Products

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

So the answer is

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