Question

Let f(x) be a polynomial. Then, the second order derivative of f(e^x) is

(a) f'' (e^x) e^2x + f'(e^x) e^x
(b) f'' (e^x) e^x + f' (e^x)
(c) f'' (e^x) e^2x + f'' (e^x) e^x
(d) f'' (e^x)

Answer 1

(a) f''(e^x)e^2x + f'(e^x)e^x

Since f(x) is a polynomial,

^{$$\begin{gathered} \therefore f^{\text{'}}\left(e^x\right)=f^{\text{'}}\left(e^x\right)e^x \\ \Rightarrow f^{\text{'}\text{'}}\left(e^x\right)=f^{\text{'}\text{'}}\left(e^x\right)(e^x{)}^2+f^{\text{'}}\left(e^x\right)e^x \\ =f^{\text{'}\text{'}}\left(e^x\right)e^{2x}+f^{\text{'}}\left(e^x\right)e^x \end{gathered}$$}