AP Calculus BC : Rules of Basic Functions: Power, Exponential Rule, Logarithmic, Trigonometric, and Inverse Trigonometric

Study concepts, example questions & explanations for AP Calculus BC

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Example Questions

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Example Question #195 : Derivative Review

Give .

Possible Answers:

Correct answer:

Explanation:

 , and the derivative of a constant is 0, so

Example Question #1 : Rules Of Basic Functions: Power, Exponential Rule, Logarithmic, Trigonometric, And Inverse Trigonometric

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 #2 : Rules Of Basic Functions: Power, Exponential Rule, Logarithmic, Trigonometric, And Inverse Trigonometric

Differentiate .

Possible Answers:

Correct answer:

Explanation:

, so

Example Question #2 : Rules Of Basic Functions: Power, Exponential Rule, Logarithmic, Trigonometric, And Inverse Trigonometric

Give the second derivative of .

Possible Answers:

Correct answer:

Explanation:

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

, so

Example Question #3 : Rules Of Basic Functions: Power, Exponential Rule, Logarithmic, Trigonometric, And Inverse Trigonometric

Give .

Possible Answers:

Correct answer:

Explanation:

, and the derivative of a constant is 0, so

Example Question #4 : Rules Of Basic Functions: Power, Exponential Rule, Logarithmic, Trigonometric, And Inverse Trigonometric

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 .

Example Question #6 : Rules Of Basic Functions: Power, Exponential Rule, Logarithmic, Trigonometric, And Inverse Trigonometric

Find the derivative of the function 

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

We can use the (first part of) the Fundemental Theorem of Calculus to "cancel out" the integral.

. Start

. Take the derivative of both sides with respect to .

To "cancel out" the integral and the derivative sign, verify that the lower bound on the integral is a constant (It's  in this case), and that the upper limit of the integral is a function of , (it's  in this case).

Afterward, plug  in for , and ultilize the Chain Rule to complete using the Fundemental Theorem of Calculus.

 

Example Question #7 : Rules Of Basic Functions: Power, Exponential Rule, Logarithmic, Trigonometric, And Inverse Trigonometric

Find the derivative of:  

Possible Answers:

Correct answer:

Explanation:

The derivative of inverse cosine is:

The derivative of cosine is:

Combine the two terms into one term.

Example Question #8 : Rules Of Basic Functions: Power, Exponential Rule, Logarithmic, Trigonometric, And Inverse Trigonometric

Find the derivative of the function 

Possible Answers:

None of the other answers

Does not exist

Correct answer:

Explanation:

To find the derivative of this function, we need to use the Fundemental Theorem of Calculus Part 1 (As opposed to the 2nd part, which is what's usually used to evaluate definite integrals)

 

. Start

. Take derivatives of both sides.

. "Cancel" the integral and the derivative. (Make sure that the upper bound on the integral is a function of , and that the lower bound is a constant before you cancel, otherwise you may need to use some manipulation of the bounds to make it so.)

Example Question #9 : 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. 

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