Question

If nth derivative of e^ax sin⁡(bx+c) cos⁡(bx+c) is, e^ax r^n sin⁡(bx+c+^nα⁄2) cos⁡(bx+c+^nα⁄2) then,

(a) r = \(\sqrt{a^2+b^2}, \alpha=tan^{-1}\frac{⁡b}{a}\)

(b) r = \(\sqrt{a^2+4b^2}, \alpha=tan^{-1}⁡\frac{2b}{a}\)

(c) r = \(\sqrt{a^2+8b^2}, \alpha=tan^{-1}⁡\frac{4b}{a}\)

(d) r = \(\sqrt{a^2+16b^2}, \alpha=tan^{-1}\frac{⁡4b}{a}\)

I had been asked this question during an online exam.

Asked question is from The nth Derivative of Some Elementary Functions in division Differential Calculus of Engineering Mathematics

Answer 1

Correct choice is (b) r = \(\sqrt{a^2+4b^2}, \alpha=tan^{-1}⁡\frac{2b}{a}\)

Explanation: y = e^ax sin⁡(bx+c) cos⁡(bx+c)

y = e^ax sin⁡2(bx+c)/2

yn = e^ax r^n sin⁡(2(bx+c+nα/2))/2

yn = e^ax r^n sin⁡(bx+c+nα/2) cos⁡(bx+c+nα/2)

where

r = \(\sqrt{a^2+4b^2}\), α = tan^-1⁡2b/a.