Differentiation Rules

Rather than computing every derivative from the limit definition, we use differentiation rules that package common patterns. The essential rules: constant $(c)' = 0$; power $(x^k)' = kx^{k-1}$; sum $(f + g)' = f' + g'$; product $(fg)' = f'g + fg'$; chain rule $(f \circ g)' = (f' \circ g) \cdot g'$.

The chain rule is the most important: if $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. In Leibniz notation, derivatives multiply like fractions. This rule will be central in Chapter 14 when we extend it to multivariate functions.

Special derivatives: $\frac{d}{dx} e^x = e^x$, $\frac{d}{dx} \sin x = \cos x$, $\frac{d}{dx} \ln x = \frac{1}{x}$ (for $x > 0$).