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

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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dessinemoie

$$\displaystyle{f{{\left({x}\right)}}}={e}^{{{\ln{{x}}}}}-{e}^{{{2}{\ln{{\left({x}^{{{2}}}\right)}}}}}{)}$$
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,
$$f'(x)=1-4x^{3}$$