Let f(x) be a polynomial in x. Then the second order derivative of $f\left(e^x\right)$ with respect to x, is

$f''\left(e^x\right) . e^x+f'\left(e^x\right)$ $f''\left(e^x\right) . e^{2 x}+f'\left(e^x\right) . e^{2 x}$ $f''\left(e^x\right) e^{2 x}$ $f''\left(e^x\right) e^{2 x}+f'\left(e^x\right) . e^x$

$f''\left(e^x\right) e^{2 x}+f'\left(e^x\right) . e^x$

Let $\phi(x)=f\left(e^x\right)$ Then, $\phi'(x)=f'\left(e^x\right) . e^x$ $\Rightarrow \phi''(x)=f''\left(e^x\right) . e^x . e^x+f'\left(e^x\right) . e^x$ $=f''\left(e^x\right) . e^{2 x}+f'\left(e^x\right) . e^x$