Question

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): x⁴(5sinx - 3cosx)

Answer 1

Let f(x)= x⁴ (5 sin x - 3 cos x) 
By Product rule,
f'(x)= x⁴ \(\frac{d}{dx}\) (5sin x-3cosx) + (5sinx-3cosx)\(\frac{d}{dx}\)(x⁴)
= x⁴[5\(\frac{d}{dx}\)(sin x) - 3\(\frac{d}{dx}\) (cos x)] + (5sinx-3cos x)\(\frac{d}{dx}\)(x⁴)
=x⁴[5cosx-3(-sinx)]+ (5 sin x - 3 cos x)(4x³)
=x³[5xcosx + 3xsinx +20sinx -12cosx]