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

Hint:

When finding the second derivative of exponential functions like \( 5^x \), it is important to apply the chain rule properly. For each option, carefully differentiate, considering constants and the behavior of exponential functions. Pay special attention to terms like \( \ln(5) \), which will affect the derivatives but not change the fundamental behavior of the exponential function itself. If the second derivative doesn't match the desired form, eliminate that option.

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:

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

Answer 2

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

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

First, calculate the first derivative:

\[ \frac{d}{dx} \left(5^x \ln(5)\right) = 5^x (\ln(5))^2 \] Now, calculate the second derivative: \[ \frac{d^2}{dx^2} \left(5^x \ln(5)\right) = 5^x (\ln(5))^3 \neq 5^x \] Thus, option (1) is not the correct answer.

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

First, calculate the first derivative: \[ \frac{d}{dx} \left(5^x (\ln(5))^2\right) = 5^x (\ln(5))^3 \] Now, calculate the second derivative: \[ \frac{d^2}{dx^2} \left(5^x (\ln(5))^2\right) = 5^x (\ln(5))^4 \neq 5^x \] Thus, option (2) is not the correct answer.

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

First, calculate the first derivative: \[ \frac{d}{dx} \left(\frac{5^x}{\ln(5)}\right) = 5^x \] Now, calculate the second derivative: \[ \frac{d^2}{dx^2} \left(\frac{5^x}{\ln(5)}\right) = 5^x \ln(5) \neq 5^x \] Thus, option (3) is not the correct answer.

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

First, calculate the first derivative: \[ \frac{d}{dx} \left(\frac{5^x}{(\ln(5))^2}\right) = \frac{5^x \ln(5)}{(\ln(5))^2} \] Now, calculate the second derivative: \[ \frac{d^2}{dx^2} \left(\frac{5^x}{(\ln(5))^2}\right) = 5^x \] Thus, the correct answer is option (4).

Conclusion: The correct answer is:

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