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Algebra of Derivatives

If fx=sin x...

Question

If $f (x) = sin x sin 2 x sin 3 x$ , find $f^{n} (x)$

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Solution

$\begin{matrix} f (x) = sin x \cdot sin 2 x sin 3 x \Rightarrow∴ y = \frac{1}{2} (2 sin x \cdot sin 2 x) sin 3 x \Rightarrow y = \frac{1}{2} [cos x \cdot sin 3 x - cos 3 x \cdot sin 3 x] \Rightarrow y = \frac{1}{4} [sin 4 x + sin 2 x - sin 6 x] [sin c + sin d = 2 sin \frac{(C + D)}{2} cos \frac{(C + D)}{2}] d i f f r e n t i a t e y w . r t o x \Rightarrow y_{x}^{1} = \frac{1}{4} [4^{x} sin (4 x + \frac{x π}{2}) - 2^{x} sin (2 x + \frac{x^{π}}{2})] t h e n, y_{x}^{1} = b^{n} sin (b x + c + \frac{x^{π}}{2}) \end{matrix}$

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