Question

Derivative of \( e^2x \) with respect to \( e^x \) is:

• A. \( e^x \)
• B. \( 2e^x \)
• C. \( 2e^2x \)
• D. \( 2e^3x \)

Hint:

Use substitution to simplify differentiation of exponential functions.

Answer 1

The Correct Option is B

Step 1: Applying the chain rule.
The derivative of \( e^{2x} \) with respect to \( x \) is calculated as:\[\fracddx e^{2x} = 2e^{2x}.\]
Step 2: Substitution of \( e^x = u \).
Let \( u = e^x \). Consequently, \( e^{2x} = (e^x)^2 = u^2 \). The derivative of \( u^2 \) with respect to \( u \) is:\[\fracddu u^2 = 2u.\]
Step 3: Determining the final result.
Substituting \( u \) back with \( e^x \), the final result is \( 2e^x \).