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Question Number 72823 by Raxreedoroid last updated on 03/Nov/19

$$\mathrm{What}\mathrm{is}\mathrm{derivative}\mathrm{for}\mathrm{this}\mathrm{function}$$$$$$$$a\times b^{x-1}\times c^{\frac{1}{2}(x-1)(x-2)}\times d^{\frac{1}{6}(x-1)(x-2)(x-3)}$$

Answered by mind is power last updated on 03/Nov/19

$$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{b}}^{\mathrm{x}-1}{\mathrm{c}}^{\frac{(\mathrm{x}-1)(\mathrm{x}-2)}{2}}.{\mathrm{d}}^{\frac{(\mathrm{x}-1)(\mathrm{x}-2)(\mathrm{x}-3)}{6}}$$$$\ln \left(\mathrm{f}\left(\mathrm{x}\right)\right)=(\mathrm{x}-1)\ln \left(\mathrm{b}\right)+\frac{(\mathrm{x}-1)(\mathrm{x}-2)}{2}\ln \left(\mathrm{c}\right)+\frac{(\mathrm{x}-1)(\mathrm{x}-2)(\mathrm{x}-3)}{6}\ln \left(\mathrm{d}\right)$$$$\Rightarrow \frac{{\mathrm{f}}^{\prime }}{\mathrm{f}}=\ln \left(\mathrm{b}\right)+\frac{(2\mathrm{x}-3)}{2}\ln \left(\mathrm{c}\right)+\left(\frac{{3\mathrm{x}}^2-12\mathrm{x}+11}{6}\right)\ln \left(\mathrm{d}\right)$$$$\Rightarrow \frac{{\mathrm{f}}^{\prime }}{\mathrm{f}}=\ln \left({\mathrm{bc}}^{\frac{2\mathrm{x}-3}{2}}{\mathrm{d}}^{\frac{{3\mathrm{x}}^2-12\mathrm{x}+11}{6}}\right)$$$$\Rightarrow {\mathrm{f}}^{\prime }={\mathrm{ab}}^{\frac{\mathrm{x}-1}{}}.{\mathrm{c}}^{\frac{(\mathrm{x}-1)(\mathrm{x}-2)}{2}}.{\mathrm{d}}^{\frac{(\mathrm{x}-1)(\mathrm{x}-2)(\mathrm{x}-3)}{6}}.\ln (\mathrm{b}.{\mathrm{c}}^{\frac{2\mathrm{x}-3}{2}}.{\mathrm{d}}^{\frac{{3\mathrm{x}}^2-12\mathrm{x}+11}{6}})$$$$$$

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