What is the derivative of e^x^2?

The Mathematician — Wed, 05 Oct 2022 13:00:02 +0000

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

Using the chain rule, we will prove that the derivative of

e^{x^2}

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*}

The derivative of

e^x

e^x

, which can be verified here. Then we have

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

Substituting everything together, we get

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

So the derivative of

e^{x^2}

2xe^{x^2}

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