Call Now to Set Up Tutoring:

(888) 888-0446

Calculus AB : Differentiate Inverse Trig Functions

Study concepts, example questions & explanations for Calculus AB

Create An Account

All Calculus AB Resources

45 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Differentiate Inverse Trig Functions

Evaluate the following derivative: $\frac{d}{dx}(2sec^{-1}x)$

Possible Answers:

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

$\frac{2}{|x| \sqrt{x^2+1}}$

$\frac{2}{|x| \sqrt{x^2-1}}$

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

Correct answer:

$\frac{2}{|x| \sqrt{x^2-1}}$

Explanation:

$\frac{d}{dx}(sec^{-1}x) = \frac{1}{|x| \sqrt{x^2-1}}$

The in the original expression is a constant and can be multiplied to the identity written above.

When dealing with the derivative of $sec^{-1}x$ , it is important to keep the standalone in the denominator in absolute value bars and to make sure there is a under the radical.

Report an Error

Example Question #2 : Differentiate Inverse Trig Functions

Let $f(x)=cot^{-1}(4x)$ . Evaluate f'(2) .

Possible Answers:

$\frac{4}{65}$

$-\frac{4}{65}$

$-\frac{1}{65}$

$\frac{1}{65}$

Correct answer:

$-\frac{4}{65}$

Explanation:

$f(x)=cot^{-1}(4x)$

First, take the derivative of the function

$f'(x)= -\frac{4}{1+(4x)^2}= -\frac{4}{1+16x^2}$

Especially when given inverse trigonometry derivative questions, be on the lookout for multiple functions embedded in the same problem. For example, in this problem there is both an outer function ( $cot^{-1}$ ) and an inner function (). Because there is more than one function, chain rule must be applied; thus, the derivative of the inner function must be multiplied to the derivative of the outer function.

To reach the final answer, f'(x) must be evaluated at x=2 .

$f'(2)= -\frac{4}{1+16(2)^2}= -\frac{4}{65}$

Report an Error

Example Question #2 : Differentiate Inverse Trig Functions

Evaluate the following derivative: $\frac{d}{dx}(sin^{-1}(3x^2))$

Possible Answers:

$\frac{x}{\sqrt{1-9x^4}}$

$\frac{1}{\sqrt{1-3x^2}}$

$\frac{6x}{\sqrt{1-9x^4}}$

$\frac{6x}{\sqrt{1-9x^2}}$

Correct answer:

$\frac{6x}{\sqrt{1-9x^4}}$

Explanation:

$\frac{d}{dx}(sin^{-1}(3x^2))=(\frac{1}{(\sqrt{1-3x^2)^2}})(6x)$

This problem requires chain rule; there is an outer function ( $sin^{-1}$ ) and an inner function ( 3x^2 ).

To simplify it further, square the 3x^2 and multiply the .

$\frac{d}{dx}(sin^{-1}(3x^2))=(\frac{1}{(\sqrt{1-3x^2)^2}})(6x) = \frac{6x}{\sqrt{1-9x^4}}$

Report an Error

Example Question #2 : Differentiate Inverse Trig Functions

Evaluate the following derivative: $\frac{d}{dx}(6cos^{-1}(2x))$

Possible Answers:

$\frac{-12}{\sqrt{1-x^2}}$

$\frac{-12}{\sqrt{1-2x^2}}$

$\frac{-12}{\sqrt{1-4x^2}}$

$\frac{12}{\sqrt{1-4x^2}}$

Correct answer:

$\frac{-12}{\sqrt{1-4x^2}}$

Explanation:

$\frac{d}{dx}(6cos^{-1}(2x))=(6)(2)(-\frac{1}{\sqrt{1-(2x)^2}})$

This problem requires chain rule, because there is an outer and inner function.

The outer function is $cos^{-1}$ and the inner is .

To fully simplify the expression, multiply the constants and square the term.

$\frac{d}{dx}(6cos^{-1}(2x))=(6)(2)(-\frac{1}{\sqrt{1-(2x)^2}}) = \frac{-12}{\sqrt{1-4x^2}}$

Report an Error

Example Question #3 : Differentiate Inverse Trig Functions

Let $f(x)=(2x)tan^{-1}(x)$ . What is f'(x) ?

Possible Answers:

$f'(x)=2tan^{-1}(x)$

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

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

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

Correct answer:

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

Explanation:

$f(x)=(2x)tan^{-1}(x)$

This is an example of when product rule must be used to find the derivative, because there are two functions ( and $tan^{-1}(x))$ being multiplied together within f(x) .

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

Report an Error

Example Question #3 : Differentiate Inverse Trig Functions

Evaluate the following derivative: $\frac{d}{dx}(csc^{-1}(10x))$

Chain rule must be used to properly find the derivative, because there is an outer function of $csc^{-1}$ and an inner function of 10x present.

$\frac{d}{dx}(csc^{-1}(10x))=(10)(-\frac{1}{|10x| \sqrt{(10x)}^2-1})=\frac{-10}{|10x| \sqrt{100x^2-1}}$

To simplify further, the 10 in the numerator cancels out the 10 in the denominator.

$\frac{d}{dx}(csc^{-1}(10x))=\frac{-10}{|10x| \sqrt{100x^2-1}}=\frac{-1}{|x| \sqrt{100x^2-1}}$

Possible Answers:

$\frac{1}{|x| \sqrt{100x^2-1}}$

$\frac{-10}{|x| \sqrt{x^2-1}}$

$\frac{-1}{|x| \sqrt{100x^2-1}}$

$\frac{-1}{|10x| \sqrt{100x^2-1}}$

Correct answer:

$\frac{-1}{|10x| \sqrt{100x^2-1}}$

Explanation:

$\frac{d}{dx}(csc^{-1}(10x))$

Report an Error

Example Question #1 : Differentiate Inverse Trig Functions

Which derivative would produce the same expression as $\frac{d}{dx}(-sec^{-1}(5x))$ ?

Possible Answers:

$\frac{d}{dx}(-tan^{-1}(5x))$

$\frac{d}{dx}(csc^{-1}(5x))$

$\frac{d}{dx}(sec^{-1}(5x))$

$\frac{d}{dx}(cot^{-1}(5x))$

Correct answer:

$\frac{d}{dx}(csc^{-1}(5x))$

Explanation:

By examining the inverse trigonometric function derivatives, it can be shown that $\frac{d}{dx}(csc^{-1}x)=\frac{d}{dx}(-sec^{-1}x)$ . The derivatives of these two inverse functions are negatives of one another.

Report an Error

Example Question #3 : Differentiate Inverse Trig Functions

Let $f(x)=(sin^{-1}x)^3$ . Find f'(x) .

Possible Answers:

$\frac{3(sin^{-1}x)^2}{\sqrt{1-x^2}}$

$\frac{3(cos^{-1}x)^2}{\sqrt{1-x^2}}$

$\frac{cos^{-1}x}{\sqrt{1-x^2}}$

$\frac{3(sin^{-1}x)}{\sqrt{1-x^2}}$

Correct answer:

$\frac{3(sin^{-1}x)^2}{\sqrt{1-x^2}}$

Explanation:

$f(x)=(sin^{-1}x)^3$

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

This question invokes the use of chain rule.

Report an Error

Example Question #1 : Differentiate Inverse Trig Functions

Let $f(x)=cos^{-1}(\sqrt{3x-1})$ . Find f'(x) .

Possible Answers:

$\frac{3}{2(\sqrt{3x-1})(\sqrt{2-3x})}$

$\frac{1}{(\sqrt{2-3x})}$

$\frac{-3}{2(\sqrt{3x-1})(\sqrt{2-3x})}$

$\frac{-3 sin ^{-1}x}{2(\sqrt{2-3x})}$

Correct answer:

$\frac{-3}{2(\sqrt{3x-1})(\sqrt{2-3x})}$

Explanation:

$f(x)=cos^{-1}(\sqrt{3x-1})$

This question requires the use of chain rule. In this case, there are three functions to consider: first, $cos^{-1}$ ; second, the square root; and third, 3x-1 . In order to evaluate, the derivative of each of the three functions must be found and multiplied together.

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

Report an Error

Example Question #1 : Differentiate Inverse Trig Functions

Evaluate the following derivative: $\frac{d}{dx}(tan^{-1}(e^{2x}))$

Possible Answers:

$\frac{2e^{2x}}{1+e^{2x}}$

$\frac{e^{2x}}{1+e^{2x}}$

$\frac{2e^{2x}}{1+e^{4x}}$

$\frac{2}{1+e^{4x}}$

Correct answer:

$\frac{2e^{2x}}{1+e^{4x}}$

Explanation:

This problem requires use of the chain rule twice; the first time addresses the inner function of $e^{2x}$ while the second accounts for the function .

$\frac{d}{dx}(tan^{-1}(e^{2x}))=\frac{1}{1+(e^{2x})^2}*e^{2x}*2=\frac{2e^{2x}}{1+e^{4x}}$

Report an Error

View Tutors

Nathan
Certified Tutor

University of the West Indies, Bachelor of Laws, Prelaw Studies.

View Tutors

Safaa
Certified Tutor

University of Montreal, Bachelor of Science, Mathematics.

View Tutors

Loveleen
Certified Tutor

Maharishi Dayanand University, Master's/Graduate, Education.

All Calculus AB Resources

45 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Computer Science Tutors in Philadelphia, Spanish Tutors in Boston, Chemistry Tutors in San Diego, Computer Science Tutors in San Francisco-Bay Area, ISEE Tutors in San Diego, Biology Tutors in Los Angeles, LSAT Tutors in Denver, Calculus Tutors in San Diego, MCAT Tutors in San Francisco-Bay Area, Biology Tutors in San Diego

Popular Courses & Classes

ISEE Courses & Classes in Denver, SSAT Courses & Classes in Los Angeles, SSAT Courses & Classes in Philadelphia, SSAT Courses & Classes in Houston, ACT Courses & Classes in New York City, LSAT Courses & Classes in Miami, MCAT Courses & Classes in Seattle, GMAT Courses & Classes in Phoenix, GRE Courses & Classes in San Diego, SAT Courses & Classes in Boston

Popular Test Prep

ISEE Test Prep in Miami, LSAT Test Prep in Boston, MCAT Test Prep in Chicago, ACT Test Prep in Washington DC, ISEE Test Prep in San Francisco-Bay Area, GMAT Test Prep in Phoenix, ISEE Test Prep in Phoenix, SAT Test Prep in Boston, SSAT Test Prep in Phoenix, GMAT Test Prep in Chicago

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

Calculus AB : Differentiate Inverse Trig Functions

Study concepts, example questions & explanations for Calculus AB

All Calculus AB Resources

Example Questions

Example Question #1 : Differentiate Inverse Trig Functions

Example Question #2 : Differentiate Inverse Trig Functions

Example Question #2 : Differentiate Inverse Trig Functions

Example Question #2 : Differentiate Inverse Trig Functions

Example Question #3 : Differentiate Inverse Trig Functions

Example Question #3 : Differentiate Inverse Trig Functions

Example Question #1 : Differentiate Inverse Trig Functions

Example Question #3 : Differentiate Inverse Trig Functions

Example Question #1 : Differentiate Inverse Trig Functions

Example Question #1 : Differentiate Inverse Trig Functions

All Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: