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

## Example Questions

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### Example Question #1 : Derivative Defined As Limit Of Difference Quotient

Evaluate .

Explanation:

To find , substitute  and use the chain rule:

So

and

### Example Question #2 : Derivative Defined As Limit Of Difference Quotient

What is the equation of the line tangent to the graph of the function

at the point  ?

Explanation:

The slope of the line tangent to the graph of  at  is

, which can be evaluated as follows:

The equation of the line with slope  through  is:

### Example Question #3 : Derivative Defined As Limit Of Difference Quotient

What is the equation of the line tangent to the graph of the function

at the point  ?

Explanation:

The slope of the line tangent to the graph of  at  is

, which can be evaluated as follows:

, the slope of the line.

The equation of the line with slope  through  is:

### Example Question #4 : Derivative Defined As Limit Of Difference Quotient

What is the equation of the line tangent to the graph of the function

at  ?

Explanation:

The slope of the line tangent to the graph of  at  is

, which can be evaluated as follows:

, the slope of the line.

The equation of the line with slope  through  is:

### Example Question #5 : Derivative Defined As Limit Of Difference Quotient

What is the equation of the line tangent to the graph of the function

at the point  ?

Explanation:

The slope of the line tangent to the graph of  at the point  is , which can be evaluated as follows:

The line with this slope through  has equation:

### Example Question #6 : Derivative Defined As Limit Of Difference Quotient

What is the equation of the line tangent to the graph of the function

at the point  ?

Explanation:

The slope of the line tangent to the graph of  at the point  is , which can be evaluated as follows:

The line with slope 28 through  has equation:

### Example Question #1 : Derivative At A Point

Given the function , find the slope of the point .

The slope cannot be determined.

Explanation:

To find the slope at a point of a function, take the derivative of the function.

The derivative of  is .

Therefore the derivative becomes,

since .

Now we substitute the given point to find the slope at that point.

### Example Question #7 : Derivative Defined As Limit Of Difference Quotient

Find the value of the following derivative at the point  :

Explanation:

To solve this problem, first we need to take the derivative of the function. It will be easier to rewrite the equation as  from here we can take the derivative and simplify to get

From here we need to evaluate at the given point . In this case, only the x value is important, so we evaluate our derivative at x=2 to get.

### Example Question #3 : Derivative At A Point

Evaluate the value of the derivative of the given function at the point :

Explanation:

To solve this problem, first we need to take the derivative of the function.

From here we need to evaluate at the given point . In this case, only the x value is important, so we evaluate our derivative at x=1 to get

.

### Example Question #8 : 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

.

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