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Mathematics Question on Limits and derivations

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): xsin nx\frac{x}{sin\,n^x}

Answer

Let f(x)= xsin nx\frac{x}{sin\,n^x}
By the quotient rule,
f'(x)= sin xddxx−xddxxsin nxxsin2nx\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 ddx\frac{d}{dx} sinnx = n sinn-1x cos x
Therefore,
f'(x)=sinnxddx−xddxsinnxsin2nx\frac{sin^nx\frac{d}{dx}-x\frac{d}{dx}sin^nx}{sin^{2n}x}
=sinn.1−x(n sinn−axcos x)sin2nx\frac{sin^n.1-x(n\,sin^{n-a}xcos\,x)}{sin^{2n}x}
=sinn−1x(sin x−nxcos x)sin2nx\frac{sin^{n-1}x(sin\,x-nxcos\,x)}{sin^{2n}x}
=sin x−nx cos xsinn+1x\frac{sin\,x-nx\,cos\,x}{sin^{n+1}x}