Question

The second order derivative of which of the following functions is \( 20^x \)?

• A. \( \frac{20^x}{(\log_e 20)^2} \)
• B. \( 20^x (\log_e 20)^2 \)
• C. \( 20^x (\log_e 20) \)
• D. \( \frac{20^x}{\log_e 20} \)

Hint:

When differentiating exponential functions with base \( a \), remember to multiply by \(\ln a\) each time. Conversely, when integrating, divide by \(\ln a\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We need to find a function \( f(x) \) such that \( f''(x) = 20^x \). Recall that \( \frac{d}{dx}(a^x) = a^x \ln a \).

Step 2: Detailed Explanation:
Let \( f(x) = \frac{20^x}{(\ln 20)^2} \).
First derivative: \( f'(x) = \frac{d}{dx} \left[ \frac{20^x}{(\ln 20)^2} \right] = \frac{20^x \ln 20}{(\ln 20)^2} = \frac{20^x}{\ln 20} \).
Second derivative: \( f''(x) = \frac{d}{dx} \left[ \frac{20^x}{\ln 20} \right] = \frac{20^x \ln 20}{\ln 20} = 20^x \).

Step 3: Final Answer:
The function is \( \frac{20^x}{(\log_e 20)^2} \).