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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)`
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Solution
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 sinn - 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)`
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