quick chain rule review: what is the derivative of e^x^2?

Explanation:

Step1: Identify inner and outer functions

Let $u = x^{2}$, outer function $y = e^{u}$.

Step2: Differentiate outer function

The derivative of $y = e^{u}$ with respect to $u$ is $\frac{dy}{du}=e^{u}$.

Step3: Differentiate inner function

The derivative of $u = x^{2}$ with respect to $x$ is $\frac{du}{dx}=2x$.

Step4: Apply chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=e^{u}$ and $\frac{du}{dx}=2x$ back in, and replace $u$ with $x^{2}$. So $\frac{dy}{dx}=e^{x^{2}}\cdot2x = 2xe^{x^{2}}$.

Answer:

$2xe^{x^{2}}$