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Let $f(x) = e^{x^2},$ then find $f''(x).$
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Given,

 $f(x)=e^{x^{2}}$

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

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

  $f'(x)=e^{x^{2}}2x$

  $f'(x)=2xe^{x^{2}}$

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

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

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

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

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

$f’(x) =$ $e^{x^{2}}(2x)$

$f’’(x) = 2 [e^{x^{2}}+$ $xe^{x^{2}}(2x) ]$

$f’’(x) = 2 [xf’(x) + f(x) ]$
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