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Question Number 83500 by 20092104 last updated on 03/Mar/20

$$\frac{d}{dx}\left({(a+bx)}^{c+dx}\right)$$

Answered by john santu last updated on 03/Mar/20

$$\mathrm{let}\mathrm{y}={(\mathrm{a}+\mathrm{bx})}^{\mathrm{c}+\mathrm{dx}}$$$$\ln \mathrm{y}=(\mathrm{c}+\mathrm{dx})\ln (\mathrm{a}+\mathrm{bx})$$$$\frac{{\mathrm{y}}^{\prime }}{\mathrm{y}}=\mathrm{d}\ln (\mathrm{a}+\mathrm{bx})+\frac{\mathrm{b}(\mathrm{c}+\mathrm{dx})}{\mathrm{a}+\mathrm{bx}}$$$${\mathrm{y}}^{\prime }=\frac{\mathrm{dy}}{\mathrm{dx}}={(\mathrm{a}+\mathrm{bx})}^{\mathrm{c}+\mathrm{dx}}[\mathrm{d}\ln (\mathrm{a}+\mathrm{bx})$$$$+\frac{\mathrm{b}(\mathrm{c}+\mathrm{dx})}{\mathrm{a}+\mathrm{bx}}]$$

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