Question

\(\frac{d}{dx} \left[ e^{x^2} \right] \)

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

Hint:

When differentiating an exponential function with a composite exponent, use the chain rule: \( \frac{d}{dx} e^{f(x)} = e^{f(x)} \cdot f'(x) \).

Answer 1

The Correct Option is C

We are asked to differentiate \( e^{x^2} \). Step 1: Apply the chain rule
Since \( x^2 \) is in the exponent, we apply the chain rule: \[ \frac{d}{dx} e^{x^2} = e^{x^2} \cdot \frac{d}{dx} \left( x^2 \right) \]
Step 2: Differentiate the exponent
The derivative of \( x^2 \) is \( 2x \), so: \[ \frac{d}{dx} e^{x^2} = 2x e^{x^2} \] Thus, the derivative of \( e^{x^2} \) is \( 2x e^{x^2} \).