StatlectThe Digital Textbook
Index > Mathematical tools > Derivatives review

Calculus - Derivatives - Review - Multiple choice test 2

Take a multiple choice test on derivatives.

Topics

In this test you will be asked questions about the following topics:

Instructions

When you give a right answer, it is marked with nav_button.

When you give a wrong answer, it is marked with nav_button and the right answer is marked with nav_button.

When you answer a question, your score is updated and you can read an explanation of the right answer before going to the next question.

Get started

Are you ready?

nav_button Start taking the test now.

Score: 0% (0 out of 0)

Question 1. Define[eq1]The derivative of $fleft( x
ight) $ is:

nav_button [eq2]

nav_button [eq3]

nav_button [eq4]

nav_button [eq5]

nav_button Explanation | nav_button Next question

Explanation. The derivative of the logarithmic function [eq6] is:[eq7]See the calculus review page entitled Derivatives - Review.

nav_button Next question

Question 2. Define[eq8]The second derivative of $fleft( x
ight) $ is:

nav_button [eq9]

nav_button [eq10]

nav_button [eq11]

nav_button [eq12]

nav_button Explanation | nav_button Next question

Explanation. The first derivative of the logarithmic function [eq13] is:[eq14]Therefore, the second derivative of $fleft( x
ight) $ is:[eq15]Using the formula for the derivative of a power function, we obtain:[eq16]See the calculus review page entitled Derivatives - Review.

nav_button Next question

Question 3. Define[eq17]The derivative of $fleft( x
ight) $ is:

nav_button [eq18]

nav_button [eq19]

nav_button [eq20]

nav_button [eq21]

nav_button Explanation | nav_button Next question

Explanation. The derivative of a generic logarithmic function [eq22] is:[eq23]Here $b=2$. Therefore:[eq24]See the calculus review page entitled Derivatives - Review.

nav_button Next question

Question 4. Define[eq25]The second derivative of $fleft( x
ight) $ is:

nav_button [eq26]

nav_button [eq27]

nav_button [eq28]

nav_button [eq29]

nav_button Explanation | nav_button Next question

Explanation. The first derivative of the logarithmic function [eq30] is:[eq31]Therefore, the second derivative of $fleft( x
ight) $ is:[eq32]Note that we have been able to use the linearity of the derivative because [eq33] is a constant. See the calculus review page entitled Derivatives - Review.

nav_button Next question

Question 5. Define[eq34]The derivative of $fleft( x
ight) $ is:

nav_button [eq35]

nav_button [eq36]

nav_button [eq37]

nav_button [eq38]

nav_button Explanation | nav_button Next question

Explanation. $sqrt{2}$ is a constant. So, by the linearity of the derivative:[eq39]Now, the rule for differentiating exponential functions is:[eq40]Therefore:[eq41]See the calculus review page entitled Derivatives - Review.

nav_button Next question

Question 6. Define[eq42]The second derivative of $fleft( x
ight) $ is:

nav_button [eq43]

nav_button [eq44]

nav_button [eq45]

nav_button [eq46]

nav_button Explanation | nav_button Next question

Explanation. The first derivative of the exponential function [eq47] is:[eq48]Therefore, the second derivative of $fleft( x
ight) $ is:[eq49]Note that we have been able to use the linearity of the derivative because [eq50] is a constant. See the calculus review page entitled Derivatives - Review.

nav_button Next question

Question 7. Define[eq51]The derivative of $fleft( x
ight) $ is:

nav_button [eq52]

nav_button [eq53]

nav_button [eq54]

nav_button [eq55]

nav_button Explanation | nav_button Next question

Explanation. By linearity of the derivative:[eq56]The first summand is: [eq57]because the derivative of a constant is 0. The second summand is:[eq58]by the rule for differentiating exponentials. Therefore:[eq59]See the calculus review page entitled Derivatives - Review.

nav_button Next question

Question 8. Define[eq60]The derivative of $fleft( x
ight) $ is:

nav_button [eq61]

nav_button [eq62]

nav_button [eq63]

nav_button [eq64]

nav_button Explanation | nav_button Next question

Explanation. The derivative of $fleft( x
ight) $ is:[eq65]Note that we have been able to use the linearity of the derivative because [eq66] is a constant. See the calculus review page entitled Derivatives - Review.

nav_button Next question

Question 9. The function:[eq67]is the derivative of:

nav_button [eq68]

nav_button [eq69]

nav_button [eq70]

nav_button [eq71]

nav_button Explanation | nav_button Next question

Explanation. By linearity:[eq72]See the calculus review page entitled Derivatives - Review.

nav_button Next question

Question 10. The function:[eq73]is the second derivative of:

nav_button [eq74]

nav_button [eq75]

nav_button [eq76]

nav_button [eq77]

nav_button Explanation

Explanation. The first derivative of the function[eq78]is:[eq79]The second derivative is:[eq80]See the calculus review page entitled Derivatives - Review.