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): (4x+5sinx)(3x+7cosx)

Answer 1

Let f(x)= \(\frac{4x+5sin\,x}{3x+7cos\,x}\)
By the quotient rule,
f'(x)= (3x+7cos x)\(\frac{d}{dx}\)(4x+5sin x)-(4x+5sin x)\(\frac{d}{dx}\)(3x+7 cos x)/(3x+7 cos x)²
f'(x)=(3x+7 cos x)[4\(\frac{d}{dx}\)(x)+5\(\frac{d}{dx}\)(sin x)] -(4x+5sin x)[3\(\frac{d}{dx}\) x + 7 \(\frac{d}{dx}\)cosx]
=\(\frac{(3x+7 cos x)(4+5 cos x)-(4x+5sin x)(3-7 sin x)}{(3x+7 cos x)^2}\)
=\(\frac{15x\,cos\,x+28cos\,x+28x\,sinx-15sin\,x+35(cos^2x+sin^2x)}{(3x+7cos\,x)^2}\)
=\(\frac{35+15x\,cos\,x+28cos\,x+28sin\,x-15sin\,x}{(3x+7cos\,x)^2}\)