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

lwfrgin 2021-06-04 Answered

Find the derivatives of the functions.
\(\displaystyle{f{{\left({x}\right)}}}={e}^{{{\ln{{x}}}}}-{e}^{{{2}{\ln{{\left({x}^{{{2}}}\right)}}}}}{}\)

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Expert Answer

dessinemoie
Answered 2021-06-05 Author has 12886 answers

\(\displaystyle{f{{\left({x}\right)}}}={e}^{{{\ln{{x}}}}}-{e}^{{{2}{\ln{{\left({x}^{{{2}}}\right)}}}}}{)}\)
Apply \(\displaystyle{{\ln{{a}}}^{{{n}}}=}{n}{\ln{{a}}}\)
\(f(x)=e^{\ln x)-e^{ln(x^{2})^{2}}}\) Apply \(\displaystyle{\left({x}^{{{m}}}\right)}^{{{n}}}={x}^{{{m}{n}}}\)
\(\displaystyle{f{{\left({x}\right)}}}={e}^{{{\ln{{x}}}}}-{e}^{{{\ln{{x}}}^{{{4}}}}}\)
Recall that \(\ln e^{z}=z\), so
\(\displaystyle{f{{\left({x}\right)}}}={x}-{x}^{{{4}}}\)
Differentiate both sides with respect to x
\(\displaystyle{f}'{\left({x}\right)}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{x}-{x}^{{{4}}}\right]}\)
Therefore,
\(f'(x)=1-4x^{3}\)

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