### All High School Math Resources

## Example Questions

### 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)?

**Possible Answers:**

undefined

**Correct answer:**

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

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct answer:**

Therefore,

for any real , so , and

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

What is ?

**Possible Answers:**

**Correct answer:**

Therefore,

for any positive , so , and

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

Find the derivative of the following function:

**Possible Answers:**

**Correct answer:**

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

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

Use the Chain Rule.

Set and substitute.

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

Find the derivative of the following function:

**Possible Answers:**

**Correct answer:**

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