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Question Number 166585 by mkam last updated on 22/Feb/22

$$ify={\left(sinx\right)}^{e^x}find\frac{d\left(e^{{sin}^{-1}y}\right)}{d\left(lnx\right)}$$

Answered by Eric002 last updated on 22/Feb/22

$$ln\left(y\right)=e^{x}ln\left(sin\left(x\right)\right)$$$$\frac{dy}{dx}=e^x{\left(sin\left(x\right)\right)}^{e^x})(cot\left(x\right)+ln\left(sin\left(x\right)\right)$$$$d\left(e^{{sin}^{-1}\left(y\right)}\right)/d\left(ln\left(x\right)\right)=\frac{x{e}^{{sin}^{-1}\left(y\right)}/\sqrt{1-y^2}}{dx}dy$$$$=e^x{\left(sin\left(x\right)\right)}^{e^x}(cot\left(x\right)+ln\left(sin\left(x\right)\right))\left(\frac{{xe}^{{sin}^1\left(y\right)}}{\sqrt{1-y^2}}\right)$$

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