Compute the derivative $f’(x)$ of the function $$f(x)=\exp\left(-\frac{1}{2\sigma^2}(x-\mu)^2\right).$$

Solution: Clearly, we have $$\left(-\frac{1}{2\sigma^2}(x-\mu)^2\right)’=-\frac{1}{2\sigma^2}2(x-\mu)=\frac{-(x-\mu)}{\sigma^2}.$$Therefore, by Chain rule (5.32), we have \begin{align*}f’(x)=&\ \exp\left(-\frac{1}{2\sigma^2}(x-\mu)^2\right)\cdot \left(-\frac{1}{2\sigma^2}(x-\mu)^2\right)’\\=&\ \frac{-(x-\mu)}{\sigma^2}\cdot\exp\left(-\frac{1}{2\sigma^2}(x-\mu)^2\right)\end{align*}