The second order derivative of which of the following functions is \( 20^x \) ?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{20^x}{(\log_e 20)^2} \)
Since \( \frac{d}{dx}\left(20^x\right)=20^x\log_e 20 \), each differentiation multiplies by \( \log_e 20 \). For Option (1):
\( \displaystyle \frac{d^2}{dx^2}\left[\frac{20^x}{(\log_e 20)^2}\right]=\frac{20^x(\log_e 20)^2}{(\log_e 20)^2} \)
\( \displaystyle =20^x \)
Why the other options are incorrect
Option (2) \( 20^x(\log_e 20)^2 \) — Incorrect. This is the second derivative of \( 20^x \); its own second derivative is \( 20^x(\log_e 20)^4 \).
Option (3) \( 20^x(\log_e 20) \) — Incorrect. Its second derivative is \( 20^x(\log_e 20)^3 \).
Option (4) \( \displaystyle\frac{20^x}{\log_e 20} \) — Incorrect. Its first derivative is \( 20^x \); its second derivative is \( 20^x\log_e 20 \).