Question

The second-order derivative of which of the following functions is $5^x$?

• A. \( 5^x \log_e 5 \)
• B. \( 5^x (\log_e 5)^2 \)
• C. \( \frac{5^x}{\log_e 5} \)
• D. \( \frac{5^x}{(\log_e 5)^2} \)

Answer 1

The Correct Option is D

We need to determine which function’s second derivative equals \(5^x\). Let us check each option.

For (1) : \(5^x \ln(5)\):

\[ \frac{d}{dx} \left(5^x \ln(5)\right) = 5^x (\ln(5))^2 \]

\[ \frac{d^2}{dx^2} \left(5^x \ln(5)\right) = 5^x (\ln(5))^3 \neq 5^x \]

For (2) : \(5^x (\ln(5))^2\):

\[ \frac{d}{dx} \left(5^x (\ln(5))^2\right) = 5^x (\ln(5))^3 \]

\[ \frac{d^2}{dx^2} \left(5^x (\ln(5))^2\right) = 5^x (\ln(5))^4 \neq 5^x \]

For (3) : \(\frac{5^x}{\ln(5)}\):

\[ \frac{d}{dx} \left(\frac{5^x}{\ln(5)}\right) = 5^x \]

\[ \frac{d^2}{dx^2} \left(\frac{5^x}{\ln(5)}\right) = 5^x \ln(5) \neq 5^x \]

For (4) : \(\frac{5^x}{(\ln(5))^2}\):

\[ \frac{d}{dx} \left(\frac{5^x}{(\ln(5))^2}\right) = \frac{5^x \ln(5)}{(\ln(5))^2} \]

\[ \frac{d^2}{dx^2} \left(\frac{5^x}{(\ln(5))^2}\right) = 5^x \]

Thus, the correct answer is:

\[ \frac{5^x}{(\ln(5))^2} \]