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

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

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