\(\displaystyle{f{{\left({x}\right)}}}={e}^{{{\ln{{x}}}}}-{e}^{{{2}{\ln{{\left({x}^{{{2}}}\right)}}}}}{)}\)

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,

\(f'(x)=1-4x^{3}\)