Question

If f(x)=cos⁻1[(1-(log x)²)/(1+(log x)²)], then the value of f'(e) is equal to

• A. 1
• B. (1)/(e)
• C. (2)/(e)
• D. (2)/(e²)

Hint:

Recognizing standard trigonometric identities can simplify inverse-trig derivatives drastically.

Answer 1

The Correct Option is C

Note that: (1-t²)/(1+t²)=cos(2tan⁻1t) Here t=log x. Hence, f(x)=cos⁻1bigl(cos(2tan⁻1(log x))bigr) =2tan⁻1(log x) Differentiate: f'(x)=(2)/(1+(log x)²)·(1)/(x) At x=e, log x=1: f'(e)=(2)/(1+1)·(1)/(e)=(2)/(e)