Question

Which of the following is correct?

(a) \(\frac{d}{dx} (sin^{-1}(⁡x))=  \frac{1}{\sqrt{1-x}}\)

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

(c) \(\frac{d}{dx} (tan^{-1}(⁡x))= \frac{1}{\sqrt{x^2+1}}\)

(d) \(\frac{d}{dx} (sin^{-1)}(⁡x))= \frac{1}{x+1} \)

The question was asked during a job interview.

My doubt is from Total Derivative topic in chapter Partial Differentiation of Engineering Mathematics

Answer 1

Correct answer is (a) \(\frac{d}{dx} (sin^{-1}(⁡x))=  \frac{1}{\sqrt{1-x}}\)

To explain: Rules for derivatives of inverse trigonometric functions are:

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

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

 \(\frac{d}{dx} (tan^{-1}(⁡x))=  \frac{1}{1+x^2}\)