Question

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

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

2021-05-19
$$\displaystyle{s}{\left({x}\right)}={\ln{{\left({x}-{8}\right)}}}^{{-{2}}}{)}$$
Simpify the logarithm first, apply $$\displaystyle{{\ln{{a}}}^{{{n}}}=}{n}{\ln{{a}}}$$, so
$$\displaystyle{s}{\left({x}\right)}=-{2}{\ln{{\left({x}-{8}\right\rbrace}}}$$
Diffetrentiate both sides with respect to x
$$\displaystyle{s}'{\left({x}\right)}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[-{2}{\ln{{\left({x}-{8}\right)}}}\right]}$$
Pull out the constant
$$\displaystyle{s}'{\left({x}\right)}=-{2}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{\ln{{\left({x}-{8}\right)}}}\right]}$$ Apply $$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{\ln}{\left|{u}\right|}\right]}={\frac{{{1}}}{{{u}}}}{\left\lbrace{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}\right.}$$
$$\displaystyle{s}'{\left({x}\right)}=-{2}{\left({\frac{{{1}}}{{{x}-{8}}}}\right)}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{x}-{8}\right]}$$
Therefore,
$$\displaystyle{s}'{\left({t}\right)}=-{2}{\left({\frac{{{1}}}{{{x}-{8}}}}\right)}{\left({1}\right)}$$
$$\displaystyle{s}'{\left({x}\right)}=-{\frac{{{2}}}{{{x}-{8}}}}$$