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

Answer: (D)

$\frac{5^x}{(\log_e 5)^2}$

Answer 2

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:

$\frac{5^x}{(\ln(5))^2}$