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Section A.1 Rules for Derivatives
Sums, differences, constant factors.
\begin{align*}
\frac{d}{dx}(k f(x)) \amp= k \frac{df}{dx}\\
\frac{d}{dx}(f(x) \pm g(x)) \amp= \frac{df}{dx} \pm \frac{dg}{dx}
\end{align*}
Products, Quotients and Compositions.
\begin{align*}
\frac{d}{dx}(f(x)g(x)) \amp= \frac{df}{dx} g(x) + f(x) \frac{dg}{dx}\\
\frac{d}{dx}(f(x)/g(x)) \amp= \frac{\ds \frac{df}{dx} g(x) - f(x) \frac{dg}{dx}}{g^2(x)}\\
\frac{d}{dx} (f(g(x)) \amp= f'(g(x))\frac{dg}{dx}\\
\text{That is, with } u = g(x), y = f(u),\\
\frac{dy}{dx} \amp = \frac{dy}{du}\frac{du}{dx}
\end{align*}
