Question

Calculate the derivatives of the functions. f(x)=(4x-1)^{-1}

Derivatives
ANSWERED
asked 2021-06-20
Calculate the derivatives of the functions.
\(\displaystyle{f{{\left({x}\right)}}}={\left({4}{x}-{1}\right)}^{{-{1}}}\)

Answers (1)

2021-06-21

\(\displaystyle{f{{\left({x}\right)}}}={4}{x}-{1}{)}^{{-{1}}}\) Differentiale both sides with respect to x
\(\displaystyle{f}'{\left({x}\right)}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{\left({4}{x}-{1}\right)}^{{-{1}}}\right]}\)
Apply the Generalized Power Rule \(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{u}^{{{n}}}\right]}'=\nu^{{{n}-{1}}}{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}\),

let \(u=4x-1\)
\(\displaystyle{f}'{\left({x}\right)}=-{\left({4}{x}-{1}\right)}^{{-{1}-{1}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{4}{x}-{1}\right]}\)
Therefore,
\(\displaystyle{f}'{\left({x}\right)}=-{\left({4}{x}-{1}\right)}^{{-{2}}}{\left({4}\right)}\)
Simplify
\(\displaystyle{f}'{\left({x}\right)}=-{\frac{{{4}}}{{{\left({4}{x}-{1}\right)}^{{{2}}}}}}\)

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