Calculus 2 : Derivative Review

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #10 : Rules Of Basic Functions: Power, Exponential Rule, Logarithmic, Trigonometric, And Inverse Trigonometric

What is the rate of change of the function  at the point ?

Possible Answers:

Correct answer:

Explanation:

The rate of change of a function at a point is the value of the derivative at that point. First, take the derivative of f(x) using the power rule for each term.

Remember that the power rule is 

, and that the derivative of a constant is zero.

Next, notice that the x-value of the point (1,6) is 1, so substitute 1 for x in the derivative.

Therefore, the rate of change of f(x) at the point (1,6) is 14. 

Example Question #101 : Derivative At A Point

Calculate the derivative of  at the point .

Possible Answers:

Correct answer:

Explanation:

There are 2 steps to solving this problem.

First, take the derivative of .

Then, replace the value of x with the given point.

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 #1 : Derivative At A Point

Evaluate the first derivative if

 and .

Possible Answers:

Correct answer:

Explanation:

First we must find the first derivative of the function.

Because the derivative of the exponential function is the exponential function itelf, or

and taking the derivative is a linear operation,

we have that

Now setting 

Thus

Example Question #181 : Functions, Graphs, And Limits

What is the derivative of (2+3cos(3x))^\pi?

Possible Answers:

3\pi(2+cos(3x))^{\pi-1}sin(3x)

-3\pi(2+cos(3x))^{\pi-1}

-3\pi(2+cos(3x))^{\pi-1}sin(3x)

-3\pi(2+cos(3x))^{\pi-1}cos(3x)

3\pi(2+cos(3x))^{\pi-1}cos(3x)

Correct answer:

-3\pi(2+cos(3x))^{\pi-1}sin(3x)

Explanation:

Need to use the power rule which states: \frac{d}{dx}u^n=nu^{n-1}\frac{du}{dx}

 

In our problem \frac{du}{dx}=-3sin(3x)

Example Question #372 : Ap Calculus Bc

Give .

Possible Answers:

Correct answer:

Explanation:

 , and the derivative of a constant is 0, so

Example Question #373 : Ap Calculus Bc

Give .

Possible Answers:

Correct answer:

Explanation:

First, find the derivative  of .

, and the derivative of a constant is 0, so

 

Now, differentiate  to get .

Example Question #374 : Ap Calculus Bc

Differentiate .

Possible Answers:

Correct answer:

Explanation:

, so

Example Question #375 : Ap Calculus Bc

Give the second derivative of .

Possible Answers:

Correct answer:

Explanation:

Find the derivative of , then find the derivative of that expression.

, so

Example Question #376 : Ap Calculus Bc

Give .

Possible Answers:

Correct answer:

Explanation:

, and the derivative of a constant is 0, so

Example Question #377 : Ap Calculus Bc

Give .

Possible Answers:

Correct answer:

Explanation:

First, find the derivative  of .

Recall that , and the derivative of a constant is 0.

 

Now, differentiate  to get .

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