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): \(\frac{x}{sin\,n^x}\)

Answer 1

Let f(x)= \(\frac{x}{sin\,n^x}\)
By the quotient rule,
f'(x)= \(\frac{sin\,x\frac{d}{dx}x-x\frac{d}{dx}\frac{x}{sin\,n^x}}{\frac{x}{sin^{2n}x}}\)
It can be easily shown that \(\frac{d}{dx}\) sinⁿx = n sin^n-1x cos x
Therefore,
f'(x)=\(\frac{sin^nx\frac{d}{dx}-x\frac{d}{dx}sin^nx}{sin^{2n}x}\)
=\(\frac{sin^n.1-x(n\,sin^{n-a}xcos\,x)}{sin^{2n}x}\)
=\(\frac{sin^{n-1}x(sin\,x-nxcos\,x)}{sin^{2n}x}\)
=\(\frac{sin\,x-nx\,cos\,x}{sin^{n+1}x}\)