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Question Number 69680 by 20190927 last updated on 26/Sep/19

$$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^{\mathrm{sinx}},0<\mathrm{x}<\frac{\pi }{2}\mathrm{find}{\mathrm{f}}^{\prime }\left(\mathrm{x}\right)$$

Answered by MJS last updated on 26/Sep/19

$$\frac{d}{dx}\left[u^v\right]=u^v(\frac{u^{\prime }v}{u}+v^{\prime }\ln u)$$ $$\Rightarrow$$ $$f^{\prime }\left(x\right)=x^{\sin x}(\frac{\sin x}{x}+\cos x\ln x)$$

Commented by20190927 last updated on 26/Sep/19

$$\mathrm{thank}\mathrm{you}\mathrm{so}\mathrm{much}$$

Answered by Henri Boucatchou last updated on 26/Sep/19

$${\mathit{f}}^{\prime }\left(\mathit{x}\right)={\left({\mathit{e}}^{\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathit{l}\mathit{n}\mathit{x}}\right)}^{\prime }={\left(\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathit{l}\mathit{n}\mathit{x}\right)}^{\prime }\mathit{f}\left(\mathit{x}\right)$$ $$=(\mathit{c}\mathit{o}\mathit{s}\mathit{x}\mathit{l}\mathit{n}\mathit{x}+\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{\mathit{x}}){\mathit{x}}^{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}$$

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