Answer:
(1) e2x [ (2x -1 )cot x – x cosec2x]/x2
(2) e2x [ (2x +1 )cot x – cosec2x]/x2
(3) e2x [ (2x -1 )cot x + cosec2x]/x2
(4) none of these
Solution:
Given y = e2x cos x /x sin x
Differentiate w.r.t.x
Use quotient rule
dy/dx = [x sin x (d/dx)e2x cos x – e2x cos x (d/dx) x sin x]/ (x sin x)2
= [x sin x (e2x ×2× cos x – e2x sin x ) – e2x cos x ( sin x+ x cos x)]/ (x sin x)2
= [ x 2 sin x cos x e2x – e2x x sin2 x – e2x cos x sin x – e2x x cos2 x]/ (x sin x)2
= [ x sin 2x e2x – x e2x(sin2x+cos2x) – e2x sin x cos x]/(x sin x)2
= [ x e2x sin 2x – x e2x – e2x sin x cos x]/(x sin x)2
= [ 2x e2x cot x – x e2x cosec2 x – e2x cot x]/x2
= e2x [ 2x cot x – x cosec2 x – cot x]/x2
= e2x [ (2x -1 )cot x – x cosec2x]/x2
Hence option (1) is the answer
