Question

Evaluate \(\frac{d[Tan^n (x)+Tanx^n+Tan^{-1} x+Tan(nx)}{dx}]\) is

(a) \(nTan^{n-1} xSec^2 x+nx^{n-1} Sec^2 x^n+1/(1+x^2)+nTan(nx)Sec^2 (nx)\)

(b) \(nTan^{n-1} xSec^2 x+nx^{n-1} Sec^2 x^n+1/(1+x^2)+nSec^2 (nx)\)

(c) \(nTan^{n-1} xSec^2 x+nx^{n-1} Sec^2 x^n+1/(1-x^2)+nSec^2 (nx)\)

(d) \(2nTan^{n-1} xSec^2 x+nx^{n-1} Sec^2 x^n+1/(1+x^2)+nSec^2 (nx)\)

This question was posed to me in an interview for internship.

The origin of the question is Limits and Derivatives of Several Variables in chapter Partial Differentiation of Engineering Mathematics

Answer 1

Correct choice is (b) \(nTan^{n-1} xSec^2 x+nx^{n-1} Sec^2 x^n+1/(1+x^2)+nSec^2 (nx)\)

Easiest explanation: \(\frac{d[Tan^n (x)+Tanx^n+Tan^{-1} x+Tan(nx)}{dx}]\)

\(=nTan^{n-1} xSec^2 x+nx^{n-1} Sec^2 x^n+\frac{1}{1+x^2}+nSec^2 (nx)\).