All AP Calculus BC Resources
Example Questions
Example Question #2 : Derivative At A Point
Evaluate the value of the derivative of the given function at the point :
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 #94 : Derivatives
Given , find the value of at the point .
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 #95 : Derivatives
Given , find the value of at the point .
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 .
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 #11 : Derivative Defined As Limit Of Difference Quotient
Find the derivative of the following function at :
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 #31 : Derivative At A Point
What is the slope of a function at the point ?
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 #121 : Derivative Review
What is the slope of the tangent line to the function
when
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 #11 : Derivative Defined As Limit Of Difference Quotient
Find for
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 #51 : Derivative At A Point
Find the second derivative of the following function at :
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 #171 : Derivative Review
Calculate the derivative of at the point .
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
Certified Tutor
Certified Tutor