Question

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

Derivatives
ANSWERED
asked 2021-05-18
Find the derivatives of the functions.
\(\displaystyle{s}{\left({x}\right)}={\ln{{\left({x}-{8}\right)}}}^{{-{2}}}{)}\)

Answers (1)

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,
\(\displaystyle{s}'{\left({t}\right)}=-{2}{\left({\frac{{{1}}}{{{x}-{8}}}}\right)}{\left({1}\right)}\)
\(\displaystyle{s}'{\left({x}\right)}=-{\frac{{{2}}}{{{x}-{8}}}}\)
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