Given,

$f(x)=e^{x^{2}}$

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

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

$f'(x)=e^{x^{2}}2x$

$f'(x)=2xe^{x^{2}}$

$f''(x)=2\frac{\mathrm{d} }{\mathrm{d} x}(xe^{x^{2}})$

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

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

$f''(x)=2xe^{x^{2}}(2x)+2e^{x^{2}}$

$f''(x)=2e^{x^{2}}(2x^{2}+1)$