Question

Nth derivative of \(\frac{1}{(1+x^2)}\)?

(a) (-1)^nn! r^-n-1 Sin(n+1)ϑ

(b) (-1)^n(n)! r^-n-1 Sin(-n-1)ϑ

(c) (-1)^n+1(n+1)! r^-n-1 Sin(n+1)ϑ

(d) (-1)^n+1(n+1)! r^-n-1 Sin(-n-1)ϑ

The question was posed to me during an interview for a job.

My question comes from The nth Derivative of Some Elementary Functions topic in section Differential Calculus of Engineering Mathematics

Answer 1

Correct choice is (a) (-1)^nn! r^-n-1 Sin(n+1)ϑ

To explain: Here,

Y = \(\frac{1}{(x+i)(x-i)}\)

Y = \(\frac{1}{2i(x-i)} – \frac{1}{2i(x+i)}\)

\(y_n = \frac{-1^n n!}{2i}[(x-i)^{-n-1}-(x+i)^{-n-1}]\)

put x=rcos ϑ and y=rsinϑ, we get

\(y_n=\frac{(-1)^n n!}{2i}[(re^{-iθ})^{-n-1}-(re^{iθ})^{-n-1}]\) (By Eulers Identity)

\(y_n=\frac{(-1)^n n!r^{-n-1}}{2i}[e^{iθ(n+1)}-e^{-iθ(n+1)}]\)

\(y_n=(-1)^n n!r^{-n-1} Sin((n+1)ϑ)\)