# AP Calculus BC : Derivative Defined as Limit of Difference Quotient

## Example Questions

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### Example Question #11 : Derivative Defined As Limit Of Difference Quotient

Given , find the value of  at the point

Explanation:

Given the function , we can use the Power Rule

for all  to find its derivative:

.

Plugging in the -value of the point  into , we get

.

### Example Question #12 : Derivative Defined As Limit Of Difference Quotient

Find the derivative of  at point .

Explanation:

Use either the FOIL method to simplify before taking the derivative or use the product rule to find the derivative of the function.

The product rule will be used for simplicity.

Substitute .

### Example Question #13 : Derivative Defined As Limit Of Difference Quotient

Find the derivative of the following function at :

Explanation:

The derivative of the function is given by the product rule:

,

Simply find the derivative of each function:

The derivatives were found using the following rules:

,

Simply evaluate each derivative and the original functions at the point given, using the above product rule.

### Example Question #14 : Derivative Defined As Limit Of Difference Quotient

What is the slope of a function  at the point ?

Explanation:

Slope is defined as the first derivative of a given function.

Since , we can use the Power Rule

for all  to determine that

.

Since we're given a point , we can use the x-coordinate  to solve for the slope at that point.

Thus,

### Example Question #15 : Derivative Defined As Limit Of Difference Quotient

What is the slope of the tangent line to the function

when

Explanation:

The slope of the tangent line to a function at a point is the value of the derivative at that point. To calculate the derivative in this problem, the product rule is necessary. Recall that the product rule states that:

.

In this example,

Therefore,

, and

At x = 1, this dervative has the value

.

### Example Question #16 : Derivative Defined As Limit Of Difference Quotient

Find  for

Explanation:

In order to find the derivative, we need to find . We can find this by remembering the product rule and knowing the derivative of natural log.

Product Rule:

.

Derivative of natural log:

Now lets apply this to our problem.

### Example Question #17 : Derivative Defined As Limit Of Difference Quotient

Find the second derivative of the following function at :

Explanation:

To find the second derivative of the function, we first must find the first derivative of the function:

The derivative was found using the following rules:

The second derivative is simply the derivative of the first derivative function, and is equal to:

One more rule used in combination with some of the ones above is:

To finish the problem, plug in x=0 into the above function to get an answer of .

### Example Question #18 : Derivative Defined As Limit Of Difference Quotient

Calculate the derivative of  at the point .

Explanation:

There are 2 steps to solving this problem.

First, take the derivative of

Then, replace the value of x with the given point and evaluate

For example, if  , then we are looking for the value of  , or the derivative of  at .

Calculate

Derivative rules that will be needed here:

• Derivative of a constant is 0. For example,
• Taking a derivative on a term, or using the power rule, can be done by doing the following:

Then, plug in the value of x and evaluate

### Example Question #19 : Derivative Defined As Limit Of Difference Quotient

If , which of the following limits equals ?