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.
The objective is to identify the function whose second derivative is \(5^x\). We will evaluate each option.
Option (1): \(5^x \ln(5)\)
First derivative: \( \frac{d}{dx} \left(5^x \ln(5)\right) = 5^x (\ln(5))^2 \)
Second derivative: \( \frac{d^2}{dx^2} \left(5^x \ln(5)\right) = 5^x (\ln(5))^3 \). This is not equal to \(5^x\).
Option (2): \(5^x (\ln(5))^2\)
First derivative: \( \frac{d}{dx} \left(5^x (\ln(5))^2\right) = 5^x (\ln(5))^3 \)
Second derivative: \( \frac{d^2}{dx^2} \left(5^x (\ln(5))^2\right) = 5^x (\ln(5))^4 \). This is not equal to \(5^x\).
Option (3): \(\frac{5^x}{\ln(5)}\)
First derivative: \( \frac{d}{dx} \left(\frac{5^x}{\ln(5)}\right) = 5^x \)
Second derivative: \( \frac{d^2}{dx^2} \left(\frac{5^x}{\ln(5)}\right) = 5^x \ln(5) \). This is not equal to \(5^x\).
Option (4): \(\frac{5^x}{(\ln(5))^2}\)
First derivative: \( \frac{d}{dx} \left(\frac{5^x}{(\ln(5))^2}\right) = \frac{5^x \ln(5)}{(\ln(5))^2} \)
Second derivative: \( \frac{d^2}{dx^2} \left(\frac{5^x}{(\ln(5))^2}\right) = 5^x \). This matches the target derivative.
Therefore, the correct function is:
\[ \frac{5^x}{(\ln(5))^2} \]