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:
Recall that for \( f(x) = a^x \), the derivative is \( f'(x) = a^x \ln(a) \). The second derivative is \( f''(x) = a^x (\ln(a))^2 \). We need to find the function whose second derivative equals \( 20^x \).

Step 2: Key Formula or Approach:
If \( f''(x) = 20^x \), then \( f(x) \) is the double integral of \( 20^x \).
Alternatively, we differentiate the options to check which one yields \( 20^x \) as the second derivative.

Step 3: Detailed Explanation:
Let \( f(x) = \frac{20^x}{(\ln 20)^2} \).
\( f'(x) = \frac{d}{dx} \left[ \frac{1}{(\ln 20)^2} \cdot 20^x \right] = \frac{1}{(\ln 20)^2} \cdot 20^x \cdot \ln 20 = \frac{20^x}{\ln 20} \).
\( f''(x) = \frac{d}{dx} \left[ \frac{1}{\ln 20} \cdot 20^x \right] = \frac{1}{\ln 20} \cdot 20^x \cdot \ln 20 = 20^x \).

Step 4: Final Answer:
The function is \( \frac{20^x}{(\log_e 20)^2} \).