Question

Find the derivative of \[ f(x)=e^{x^2} \]

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

Hint:

Whenever an exponential contains another function inside it, always apply the chain rule.

Answer 1

The Correct Option is B

Concept: Use Chain Rule: \[ \frac{d}{dx}(e^u)=e^u\frac{du}{dx} \] where \(u\) is a function of \(x\).

Step 1: Choose inner function Let: \[ u=x^2 \] Differentiate: \[ \frac{du}{dx}=2x \]

Step 2: Apply chain rule \[ \frac{d}{dx}(e^{x^2}) = e^{x^2}\cdot\frac{d}{dx}(x^2) \] \[ = e^{x^2}\cdot2x \] \[ = 2xe^{x^2} \]

Step 3: Write final derivative Hence: \[ \boxed{2xe^{x^2}} \] Final Answer: \[ \boxed{2xe^{x^2}} \]