Differentiation rules

Let $I\subseteq\mathbb R$ be an interval, let $g:I\to J$ and $f:J\to\mathbb R$ be functions on intervals, and fix an interior point $x\in I$. Assume $f$ and $g$ are differentiable at the points where they are used. Then:

• Linearity. For $c\in\mathbb R$,
  $$(f+g)'(x)=f'(x)+g'(x),\qquad (cf)'(x)=c\,f'(x).$$
• Product rule.
  $$(fg)'(x)=f'(x)g(x)+f(x)g'(x).$$

• Quotient rule. If $g(x)\neq 0$, then
  $$\left(\frac{f}{g}\right)'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{(g(x))^2}.$$

• Chain rule. If $g$ is differentiable at $x$ and $f$ is differentiable at $g(x)$, then
  $$(f\circ g)'(x)=f'(g(x))\,g'(x).$$