Question

The derivative of \( 2^x \) with respect to \( x \) is:

• A. \( \ln(x) \cdot 2^x \)
• B. \( x \cdot 2^{x-1} \)
• C. \( \ln(2) \cdot 2^x \)
• D. \( 2 \cdot 2^{x-1} \)

Hint:

The derivative of an exponential function \( a^x \) is \( \ln(a) \cdot a^x \), where \( \ln(a) \) is the natural logarithm of the base \( a \).

Answer 1

The Correct Option is C

Step 1: Understanding the derivative of exponential functions.
The derivative of \( a^x \), where \( a \) is a constant, is given by the formula \( \frac{d}{dx}(a^x) = \ln(a) \cdot a^x \). In this case, \( a = 2 \), so the derivative of \( 2^x \) is \( \ln(2) \cdot 2^x \).

Step 2: Analyzing the options.
- (A) \( \ln(x) \cdot 2^x \): This is incorrect because the derivative involves \( \ln(2) \), not \( \ln(x) \).
- (B) \( x \cdot 2^{x-1} \): This is incorrect. The correct formula for the derivative does not involve multiplying by \( x \).
- (C) \( \ln(2) \cdot 2^x \): Correct — This is the correct derivative of \( 2^x \) with respect to \( x \).
- (D) \( 2 \cdot 2^{x-1} \): This is incorrect. It does not represent the derivative of \( 2^x \).

Step 3: Conclusion.
The correct answer is (C) because the derivative of \( 2^x \) with respect to \( x \) is \( \ln(2) \cdot 2^x \).