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]
=$\frac{(3x+7 cos x)(4+5 cos x)-(4x+5sin x)(3-7 sin x)}{(3x+7 cos x)^2}$
=$\frac{15x\,cos\,x+28cos\,x+28x\,sinx-15sin\,x+35(cos^2x+sin^2x)}{(3x+7cos\,x)^2}$
=$\frac{35+15x\,cos\,x+28cos\,x+28sin\,x-15sin\,x}{(3x+7cos\,x)^2}$