Question

If $f\left(x\right)=\sin x\sin 2x\sin 3x$, find $f^n\left(x\right)$

Answer 1

$\begin{array}{c}f\left(x\right)=\sin x\cdot \sin 2x\sin 3x\\ \Rightarrow \therefore y=\frac{1}{2}\left(2\sin x\cdot \sin 2x\right)\sin 3x\\ \Rightarrow y=\frac{1}{2}\left[\cos x\cdot \sin 3x-\cos 3x\cdot \sin 3x\right]\\ \Rightarrow y=\frac{1}{4}\left[\sin 4x+\sin 2x-\sin 6x\right]\\ \left[\sin c+\sin d=2\sin \frac{\left(C+D\right)}{2}\cos \frac{\left(C+D\right)}{2}\right]\\ diffrentiateyw.rtox\\ \Rightarrow y_x^1=\frac{1}{4}\left[4^x\sin \left(4x+\frac{x\pi }{2}\right)-2^x\sin \left(2x+\frac{x^{\pi }}{2}\right)\right]\\ then,y_x^1=b^n\sin \left(bx+c+\frac{x^{\pi }}{2}\right)\end{array}$