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Question Number 83257 by 09658867628 last updated on 29/Feb/20

$$findthederivtiveofy=\frac{{10}^x}{{log}_{10}x}$$

Answered by TANMAY PANACEA last updated on 29/Feb/20

$$y=\frac{u}{v}\to \frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$$$$u={10}^x\to lnu=xln10$$$$\frac{1}{u}\frac{du}{dx}=ln10\mathit{s}\mathit{o}\frac{\mathit{d}\mathit{u}}{\mathit{d}\mathit{x}}={10}^{\mathit{x}}\mathit{l}\mathit{n}10$$$$\mathit{v}={\mathit{l}\mathit{o}\mathit{g}}_{10}\mathit{x}\mathit{s}\mathit{o}\mathit{v}=\frac{\mathit{l}\mathit{n}\mathit{x}}{\mathit{l}\mathit{n}10}$$$$\frac{\mathit{d}\mathit{v}}{\mathit{d}\mathit{x}}=\frac{1}{\mathit{x}\mathit{l}\mathit{n}10}$$$$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=\frac{{\mathit{l}\mathit{o}\mathit{g}}_{10}\mathit{x}.{10}^{\mathit{x}}\mathit{l}\mathit{n}10-{10}^{\mathit{x}}.\frac{1}{\mathit{x}\mathit{l}\mathit{n}10}}{{\left({\mathit{l}\mathit{o}\mathit{g}}_{10}\mathit{x}\right)}^2}$$

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