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: Chain rule
The derivative of \( e^2x \) with respect to \( x \) is: \[ \fracddx e^2x = 2e^2x. \]
Step 2: Substitute \( e^x = u \)
Let \( u = e^x \), so \( e^2x = u^2 \). Differentiate \( u^2 \) with respect to \( u \): \[ \fracddu u^2 = 2u. \]
Step 3: Final result
Since \( u = e^x \), the result is \( 2e^x \).