Question

If \( y = e^{-2x} \), then \( \frac{d^3y}{dx^3} \) is equal to:

• A. \( 2e^{-2x} \)
• B. \( e^{-4x} \)
• C. \( 4e^{-4x} \)
• D. \( -8e^{-2x} \)

Hint:

When differentiating exponential functions of the form \( e^{ax} \), apply the chain rule repeatedly.

Answer 1

The Correct Option is D

Step 1: Differentiate \( y = e^{-2x} \): \[ \frac{dy}{dx} = -2e^{-2x}. \] Step 2: Differentiate again: \[ \frac{d^2y}{dx^2} = 4e^{-2x}. \] Step 3: Differentiate once more: \[ \frac{d^3y}{dx^3} = -8e^{-2x}. \]