We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

High School Math : Finding Derivatives

Study concepts, example questions & explanations for High School Math

Create An Account

All High School Math Resources

8 Diagnostic Tests 613 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #71 : Calculus I — Derivatives

Find the derivative $y^{'}$ for $y = x^{2}2^x$

Possible Answers:

$x2^{x+1}+\ln(2)x^{2}2^{x}$

$2x2^{x}-x^22^{x-1}$

$2x2^{x}+x^{3}2^{x-1}$

$2x\ln(2)2^{x}$

Correct answer:

$x2^{x+1}+\ln(2)x^{2}2^{x}$

Explanation:

The derivative must be computed using the product rule. Because the derivative of x^2 brings a down as a coefficient, it can be combined with 2^x to give $2^{x+1}$

Report an Error

Example Question #72 : Calculus I — Derivatives

Give the instantaneous rate of change of the function $f (x) = \left ( \frac{1}{3} \right ) ^{x}$ at x = 3 .

Possible Answers:

$-\frac{ \ln 3}{27}$

$-\frac{ 1}{27}$

$-27\ln 3$

$\frac{ \ln 3}{27}$

$\frac{ 1}{27}$

Correct answer:

$-\frac{ \ln 3}{27}$

Explanation:

The instantaneous rate of change of at $x = x _{0}$ is $f ' (x _{0})$ , so we will find f ' (x) and evaluate it at x = 3 .

$\frac{\mathrm{d}}{\mathrm{d} x} \left ( a ^{x } \right ) = \ln a \cdot a ^{x }$ for any positive , so

$f'(x) =\frac{\mathrm{d}}{\mathrm{d} x}\left [ \left ( \frac{1}{3} \right ) ^{x }\right ] = \ln \frac{1}{3} \cdot \left ( \frac{1}{3} \right) ^{x }= -\frac{ \ln 3}{3^{x}}$

$f'(3) =-\frac{ \ln 3}{3^{3}} = -\frac{ \ln 3}{27}$

Report an Error

Example Question #73 : Calculus I — Derivatives

$f(x) = 5 ^{x + 1}$

What is f'(x) ?

Possible Answers:

$f'(x) = 5 ^{x}$

$f'(x) = \frac{5 ^{x+1}}{\ln 5}$

$f'(x) = \frac{5 ^{x}}{\ln 5}$

$f'(x) = \ln 5 \cdot 5 ^{x+1}$

$f'(x) = \ln 5 \cdot 5 ^{x}$

Correct answer:

$f'(x) = \ln 5 \cdot 5 ^{x+1}$

Explanation:

$f(x) = 5 ^{x + 1} = 5 ^{1} \cdot 5 ^{x } = 5 \cdot 5 ^{x }$

Therefore,

$f'(x) = 5 \cdot \frac{\mathrm{d} }{\mathrm{d} x} \left ( 5 ^{x } \right )$

$\frac{\mathrm{d}}{\mathrm{d} x} \left ( a ^{x } \right ) = \ln a \cdot a ^{x }$ for any real , so $\frac{\mathrm{d} }{\mathrm{d} x} \left ( 5 ^{x } \right ) = \ln 5 \cdot 5 ^{x}$ , and

$f'(x) = 5 \ln 5 \cdot 5 ^{x} = \ln 5 \cdot 5 \cdot 5 ^{x} = \ln 5 \cdot 5 ^{x+1}$

Report an Error

Example Question #74 : Calculus I — Derivatives

$f(x) = 6 ^{x - 2}$

What is f'(4) ?

Possible Answers:

$6 \ln 6$

$\frac{\ln 6}{6}$

$36 \ln 6$

$\ln 36$

$\ln 6$

Correct answer:

$36 \ln 6$

Explanation:

$f(x) = 6 ^{x -2} = 6 ^{-2} \cdot 6 ^{x } = \frac{1}{36} \cdot 6 ^{x }$

Therefore,

$f'(x) = \frac{1}{36} \cdot \frac{\mathrm{d} }{\mathrm{d} x} \left ( 6 ^{x } \right )$

$\frac{\mathrm{d}}{\mathrm{d} x} \left ( a ^{x } \right ) = \ln a \cdot a ^{x }$ for any positive , so $\frac{\mathrm{d} }{\mathrm{d} x} \left ( 6 ^{x } \right ) = \ln 6 \cdot 6 ^{x}$ , and

$f'(x) = \frac{1}{36} \ln 6 \cdot 6 ^{x} = \ln 6 \cdot \frac{1}{36} \cdot 6 ^{x}$

$f'(x) = \ln 6 \cdot 6 ^{x-2}$

$f'(4) = \ln 6 \cdot 6 ^{4-2} = \ln 6 \cdot 6 ^{2} = 36 \ln 6$

Report an Error

Example Question #74 : Calculus I — Derivatives

Find the derivative of the following function:

$f(x) = 3 \;\textup{sin}(x)$

Possible Answers:

$f(x) = 3 \;\textup{csc}(x)$

$f(x) = -3 \;\textup{sin}(x)$

$f(x) = -3 \;\textup{cos}(x)$

$f(x) = 3 \;\textup{cos}(x)$

$f(x) = 3 \;\textup{tan}(x)$

Correct answer:

$f(x) = 3 \;\textup{cos}(x)$

Explanation:

The derivative of $\sin (x)$ is $\cos (x)$ . It is probably best to memorize this fact (the proof follows from the difference quotient definition of a derivative).

Our function

$f(x) = 3 \;\textup{sin}(x)$

the factor of 3 does not change when we differentiate, therefore the answer is

$f(x) = 3 \;\textup{cos}(x)$

Report an Error

Example Question #2 : Specific Derivatives

$\frac{\mathrm{d}}{\mathrm{d} x}\sin(x)=?$

Possible Answers:

$\sec(x)$

$\csc(x)$

$\cos(x)$

$\sin(x)$

$-\sin(x)$

Correct answer:

$\cos(x)$

Explanation:

The derivative of a sine function does not follow the power rule. It is one that should be memorized.

$\frac{\mathrm{d} }{\mathrm{d} x}\sin(x)=\cos(x)$ .

Report an Error

Example Question #3 : Specific Derivatives

What is the second derivative of $\sin(x)$ ?

Possible Answers:

$\csc(x)$

$\cos(x)$

$\sec(x)$

$-\cos(x)$

$-\sin(x)$

Correct answer:

$-\sin(x)$

Explanation:

The derivatives of trig functions must be memorized. The first derivative is:

$\frac{\mathrm{d} }{\mathrm{d} x}\sin(x)=\cos(x)$ .

To find the second derivative, we take the derivative of our result.

$\frac{\mathrm{d} }{\mathrm{d} x}\cos(x)=-\sin(x)$ .

Therefore, the second derivative will be $-\sin(x)$ .

Report an Error

Example Question #73 : Calculus I — Derivatives

Compute the derivative of the function $f(x) = \tan x^{2}$ .

Possible Answers:

$f' (x) = x^{2} \sec^{2} x^{2}$

$f' (x) = x^{2} \sec 2x$

$f' (x) = 2x \sec^{2} x^{2}$

$f' (x) = 2x \sec x^{2}$

$f' (x) = x^{2} \tan x^{2} + 2x \sec^{2} x^{2}$

Correct answer:

$f' (x) = 2x \sec^{2} x^{2}$

Explanation:

Use the Chain Rule.

Set $u = x^{2}$ and substitute.

$f(x) = \tan u$

$\frac{\mathrm{d} f }{\mathrm{d} u} = \frac{\mathrm{d} }{\mathrm{d} u} \tan u = \sec^{2} u$

$\frac{\mathrm{d} u }{\mathrm{d} x} = \frac{\mathrm{d} }{\mathrm{d} x} x^{2} = 2x^{2-1} =2x$

$\frac{\mathrm{d} f }{\mathrm{d} x} = \frac{\mathrm{d} f }{\mathrm{d} u} \cdot \frac{\mathrm{d} u }{\mathrm{d} x}= \sec^{2} u \cdot 2x = 2x \sec^{2} x^{2}$

Report an Error

Example Question #5 : Specific Derivatives

Find the derivative of the following function:

$f(x) = x^3 + \frac{1}{x}$

Possible Answers:

f'(x) = 3x^2+x

$f'(x) = 3x^2-\frac{1}{x^2}$

$f'(x) = 3x^2+\frac{1}{x^2}$

$f'(x) = 3x^2-\frac{1}{x}$

f'(x) = 3x^2

Correct answer:

$f'(x) = 3x^2-\frac{1}{x^2}$

Explanation:

Since this function is a polynomial, we take the derivative of each term separately.

From the power rule, the derivative of

x^3

is simply

3x^2

We can rewrite $\frac{1}{x}$ as

$x^{-1}$

and using the power rule again, we get a derivative of

$-x^{-2}$ or $-\frac{1}{x^2}$

So the answer is

$f'(x) = 3x^2-\frac{1}{x^2}$

Report an Error

Example Question #11 : Specific Derivatives

What is $\frac{\mathrm{d} }{\mathrm{d} x}f(x)*g(x)\mathrm{d x} ?$

Possible Answers:

f(x)*f'(x)+g(x)*g'(x)

$\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$

f(x)g(x)+f'(x)g'(x)

f(x)g'(x)-f'(x)g(x)

f(x)g'(x)+f'(x)g(x)

Correct answer:

f(x)g'(x)+f'(x)g(x)

Explanation:

The chain rule is "first times the derivative of the second plus second times derivative of the first".

In this case, that means $\frac{\mathrm{d} }{\mathrm{d} x}f(x)*g(x)\mathrm{d x} =f(x)g'(x)+f'(x)g(x)$ .

Report an Error

View Tutors

Joshua
Certified Tutor

Murray State University, Bachelor in Arts, Spanish. Lipscomb University, Bachelor of Science, Teacher Assistant.

View Tutors

Kaylee
Certified Tutor

Brigham Young University, Bachelor, MFA - Animation.

View Tutors

Lydia
Certified Tutor

The Richard Stockton College of New Jersey, Bachelor of Science, Biochemistry and Molecular Biology.

All High School Math Resources

8 Diagnostic Tests 613 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Physics Tutors in Houston, English Tutors in New York City, Algebra Tutors in Dallas Fort Worth, Biology Tutors in Boston, Math Tutors in Phoenix, ISEE Tutors in Miami, Calculus Tutors in Seattle, ACT Tutors in Miami, Calculus Tutors in New York City, SSAT Tutors in Dallas Fort Worth

Popular Courses & Classes

GMAT Courses & Classes in Dallas Fort Worth, ISEE Courses & Classes in Washington DC, Spanish Courses & Classes in Chicago, GMAT Courses & Classes in San Francisco-Bay Area, SAT Courses & Classes in Washington DC, ACT Courses & Classes in Miami, Spanish Courses & Classes in Washington DC, ACT Courses & Classes in Boston, SSAT Courses & Classes in San Francisco-Bay Area, GMAT Courses & Classes in Atlanta

Popular Test Prep

SAT Test Prep in Atlanta, GRE Test Prep in Dallas Fort Worth, GRE Test Prep in Phoenix, MCAT Test Prep in Los Angeles, ISEE Test Prep in Phoenix, SSAT Test Prep in San Francisco-Bay Area, MCAT Test Prep in Philadelphia, ISEE Test Prep in Chicago, GRE Test Prep in Atlanta, MCAT Test Prep in Washington DC

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

High School Math : Finding Derivatives

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #71 : Calculus I — Derivatives

Example Question #72 : Calculus I — Derivatives

Example Question #73 : Calculus I — Derivatives

Example Question #74 : Calculus I — Derivatives

Example Question #74 : Calculus I — Derivatives

Example Question #2 : Specific Derivatives

Example Question #3 : Specific Derivatives

Example Question #73 : Calculus I — Derivatives

Example Question #5 : Specific Derivatives

Example Question #11 : Specific Derivatives

All High School Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: