\({f}'{\left({x}\right)}={e}^{{\frac{{x}^{2}}{{{x}^{3}+{2}}}}}{\left(\frac{{{\left({x}^{3}+{2}\right)}{\left({2}{x}\right)}-{x}^{2}{\left({3}{x}^{2}\right)}}}{{{\left({x}^{3}+{2}\right)}^{2}}}\right)}\)

\(={e}^{{\frac{{x}^{2}}{{{x}^{3}+{2}}}}}{\left(\frac{{{4}{x}-{x}^{4}}}{{{\left({x}^{3}+{2}\right)}^{2}}}\right)}\)

\(={e}^{{\frac{{x}^{2}}{{{x}^{3}+{2}}}}}{\left(\frac{{{4}{x}-{x}^{4}}}{{{\left({x}^{3}+{2}\right)}^{2}}}\right)}\)