$\large f(x)=e^{x^{2}}\ln x$
$\large f'(x)=e^{x^{2}}\frac{\mathrm{d} }{\mathrm{d} x}\ln x+\ln x \frac{\mathrm{d} }{\mathrm{d} x}e^{x^{2}}$
$\large f'(x)=e^{x^{2}}\frac{1}{x}+\ln x e^{x^{2}}\frac{\mathrm{d} }{\mathrm{d} x}x^{2}$
$\large f'(x)=e^{x^{2}}\frac{1}{x}+\ln x e^{x^{2}}(2x)$
$\large f'(x)=e^{x^{2}}(\frac{1}{x}+2x\ln x)$