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Compute the derivative of the given function

by RoRi

Compute the derivative of the function

    \[ f(x) = \sin^n x \cdot cos (nx). \]

Using the chain rule we compute,

    \begin{align*} f'(x) &= n (\sin^{n-1} x) \cos x \cos (nx) + \sin^n x (- \sin nx)(n) \\ &= (n \sin^{n-1} x)( \cos x \cos (nx) - \sin x \sin (nx)) \\ &= (n \sin^{n-1} x)\cos ((n+1)x). \end{align*}

The final equality follows from the identity \cos (x+y) = \cos x \cos y - \sin x \sin y since

    \[ \cos ((n+1)x) = \cos (x + nx) = \cos x \cos (nx) - \sin x \sin (nx). \]

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