Question

Differentiate the following w.r.t.x. :

y = `(x^2 sin x)/(x + cos x)`

Answer 1

Let y = `(x^2 sin x)/(x + cos x)`

∴ `("d"y)/("d"x) = "d"/("d"x) [(x^2 sin x)/(x + cos x)]`

= `((x + cos x) "d"/("d"x) (x^2 sin x) - (x^2 sin x) "d"/("dx") (x + cos x))/(x + cos x)^2`

= `((x + cos x) [x^2 "d"/("d"x) (sin x) + (sin x) "d"/("d"x) (x^2)] - (x^2 sin x) (1 - sin x))/(x + cos x)^2`

= `((x + cos x) [x^2 (cos x) + (sin x)(2x)] - x^2 sin x + x^2 sin^2x)/(x + cos x)^2`

= `(x^3 cos x + 2x^2 sin x + x^2 cos^2x + 2x sin x cos x + x^2 sin^2x - x^2 sinx)/(x + cos x)^2`

= `(x^3 cos x + x^2 sin x + x^2 cos^2x + x^2 sin^2x + x sin 2x)/(x + cos x)^2`

= `(x^2[x cos x + sin x + (cos^2x + sin^2x)] + x sin 2x)/(x + cos x)^2`

= `(x^2[x cos x + sin x + 1] + x sin 2x)/(x + cos x)^2`