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

Options

A. `f''(e^(x)).e^(x)+f'(e^(x))`
B. `f''(e^(x)).e^(2x)+f'(e^(x)).e^(2x)`
C. `f''(e^(x))e^(2x)`
D. `f''(e^(x))e^(2x)+f'(e^(x)).e^(x)`

Answer

D

Explanation

Let `phi(x)=f(e^(x)).` Then,
`phi'(x)=f'(e^(x)).e^(x)`
`implies" "phi''(x)=f''(e^(x)).e^(x).e^(x)+f'(e^(x)).e^(x)`
`=f''(e^(x)).e^(2x)+f'(e^(x)).e^(x)`