The second order derivative of which of the following functions is \(5^x\)?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{5^x}{\left(\log_e 5\right)^2}\)
Since \(\frac{d}{dx}\left(5^x\right) = 5^x\log_e 5\), each differentiation multiplies by \(\log_e 5\). Let \(f(x) = \frac{5^x}{(\log_e 5)^2}\):
\( \displaystyle f'(x) = \frac{5^x\log_e 5}{(\log_e 5)^2} = \frac{5^x}{\log_e 5} \)
\( \displaystyle f''(x) = \frac{5^x\log_e 5}{\log_e 5} = 5^x \)
Why the other options are incorrect
Option (1) \(5^x\log_e 5\) — Incorrect. Two more factors of \(\log_e 5\) give the second derivative \(5^x(\log_e 5)^3\).
Option (2) \(5^x\left(\log_e 5\right)^2\) — Incorrect. Its second derivative is \(5^x(\log_e 5)^4\).
Option (3) \(\displaystyle\frac{5^x}{\log_e 5}\) — Incorrect. Its first derivative is \(5^x\), so its second derivative is \(5^x\log_e 5\).