Question

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

`x/(sin^n x)`

Answer 1

Let f(x) = `x/(sin^n x)`

By quotient rule,

f'(x) = `(sin^n x d/dx x - x d/dx sin^n x)/(sin^(2n)x)`

It is shown that `d/dx sin^n x` = n sin^{n - 1} x cos x

Therefore,

f'(x) = `(sin^n x d/dx x - x d/dx sin^n x)/(sin^(2n)x)`

= `(sin^n x.1 - x (n sin^(n - 1)x cos x))/(sin^(n + 1) x)`

= `(sin^(n - 1) x(sin x - nx cosx))/(sin^(2n)x)`

= `(sin x - nx cos x)/(sin^(n + 1)x)`