Question

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

Options

• f'' (e^x) e^2x + f'(e^x) e^x

•  f'' (e^x) e^x + f' (e^x)

• f'' (e^x) e^2x + f'' (e^x) e^x

•  f'' (e^x)

Answer 1

(a) f''(e^x)e^2x + f'(e^x)e^x

Since f(x) is a polynomial,

\[\therefore f^{'} \left( e^x \right) = f^{'} \left( e^x \right) e^x \]

\[ \Rightarrow f^{''} \left( e^x \right) = f^{''} \left( e^x \right) ( e^x )^2 + f^{'} \left( e^x \right) e^x \]

\[ = f^{''} \left( e^x \right) e^{2x} + f^{'} \left( e^x \right) e^x\]