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Question Number 124845 by liberty last updated on 06/Dec/20

$$f\left(x\right)=\sin \left(\frac{x}{x-\sin \left(\frac{x}{x-\sin x}\right)}\right)$$$$\frac{df\left(x\right)}{dx}?$$

Answered by bemath last updated on 06/Dec/20

$$f{}^{\prime }\left(x\right)=\frac{1(x-\sin \left(\frac{x}{x-\sin x}\right))-x(1-\left(\frac{x-\sin x-x(1-\cos x)}{{(x-\sin x)}^2}\right)\cos \left(\frac{x}{x-\sin x}\right)}{{(x-\sin \left(\frac{x}{x-\sin x}\right))}^2}.\cos \left(\frac{x}{x-\sin \left(\frac{x}{x-\sin x}\right)}\right)$$

Answered by mindispower last updated on 06/Dec/20

$$g\left(x\right)=\frac{x}{x-sin\left(x\right)}$$$$arcsin\left(fx\right)=\frac{x}{x-sin\left(g\left(x\right)\right)}$$$$\Rightarrow \frac{f^{\prime }}{\sqrt{1-f^2}}=\frac{-sin\left(g\left(x\right)\right)+g^{\prime }cos\left(g\left(x\right)\right)}{x-sin{\left(g\left(x\right)\right)}^2}{}_{=A}$$$$justsamasAwitheg=x,g^{\prime }=1$$$$g^{\prime }=\frac{-sin\left(x\right)+cos\left(x\right)}{{(x-sin\left(x\right))}^2}$$

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