djeljenike

2021-05-18

Find the derivatives of the functions.
$s\left(x\right)={\mathrm{ln}\left(x-8\right)}^{-2}\right)$

Yusuf Keller

$s\left(x\right)={\mathrm{ln}\left(x-8\right)}^{-2}\right)$
Simpify the logarithm first, apply ${\mathrm{ln}a}^{n}=n\mathrm{ln}a$, so
$s\left(x\right)=-2\mathrm{ln}\left(x-8\right\}$
Diffetrentiate both sides with respect to x
${s}^{\prime }\left(x\right)=\frac{d}{dx}\left[-2\mathrm{ln}\left(x-8\right)\right]$
Pull out the constant
${s}^{\prime }\left(x\right)=-2\frac{d}{dx}\left[\mathrm{ln}\left(x-8\right)\right]$Apply $\frac{d}{dx}\left[\mathrm{ln}|u|\right]=\frac{1}{u}\left\{\frac{du}{dx}$
${s}^{\prime }\left(x\right)=-2\left(\frac{1}{x-8}\right)\frac{d}{dx}\left[x-8\right]$
Therefore,
${s}^{\prime }\left(t\right)=-2\left(\frac{1}{x-8}\right)\left(1\right)$
${s}^{\prime }\left(x\right)=-\frac{2}{x-8}$

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