Q. $\frac{d^{n}}{dx^{n}}(\log{x)\ }$is equal to
A) $\frac{(n - 1)!}{x^{n}}$
B) $\frac{n!}{x^{n}}$
C) $\frac{(n - 2)!}{x^{n}}$
D) $( - 1)^{n - 1}\frac{(n - 1)!}{x^{n}}$
Explanation
(d)
Let $y = \log x$
On differentiating w.r.t. $x\$from 1 to $n$ times, we get
$y_{1} = \frac{1}{x},y_{2} = - \frac{1}{x^{2}},\ y_{3} = \frac{2}{x^{3}},y_{4} = - \frac{6}{x^{4}},\ldots,$
$\therefore$ by symmctry.
$y_{n} = \frac{( - 1)^{n - 1}(n - 1)!}{x^{n}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.