**What is the derivative of e^{x^2}?**

Using the chain rule, we will prove that the derivative of e^{x^2} is 2xe^{x^2}.

**Proof.**Let h(x) = e^{x^2}, f(u) = e^u and g(x) = x^2. We will use the chain rule to determine the derivative, i.e.:

\begin{align*} h'(x) = \frac{d}{dx}f(g(x)) = f'(g(x))g'(x). \end{align*}

\begin{align*} f'(u) = e^u \quad \text{and} \quad g'(x) = 2x. \end{align*}

\begin{align*} h'(x) &= f'(g(x))g'(x) \\ &= e^{x^2} \cdot 2x \\ &= 2xe^{x^2}. \end{align*}