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Question Number 178847 by zaheen last updated on 22/Oct/22

$$y=x\sin \left(x\right)$$$$\frac{d^{35}y}{{dx}^{35}}=?$$

Answered by mr W last updated on 22/Oct/22

$$y=f\left(x\right)g\left(x\right)with$$$$f\left(x\right)=x,g\left(x\right)=\sin \left(x\right)$$$$f^{(0}\left(x\right)=x,f^{\left(1\right)}\left(x\right)=1,f^{\left(k\right)}\left(x\right)=0fork⩾2$$$$g^{(0}\left(x\right)=\sin x,g^{\left(1\right)}\left(x\right)=\cos x,g^{\left(2\right)}\left(x\right)=-\sin x$$$$g^{(2k+1)}\left(x\right)={(-1)}^k\cos x,g^{\left(2k\right)}\left(x\right)={(-1)}^k\sin x$$$$y^{\left(35\right)}=f^{\left(0\right)}\left(x\right)g^{\left(35\right)}\left(x\right)+35f^{\left(1\right)}\left(x\right)g^{\left(34\right)}\left(x\right)+0$$$$=x(-\cos x)+35\left(1\right)(-\sin x)$$$$=-x\cos x-35\sin x$$

Commented by Tawa11 last updated on 23/Oct/22

$$\mathrm{Great}\mathrm{sir}$$

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