Differentiation rules

Differentiation rules: Let $I\subseteq\mathbb{R}$ be an interval , and let $f,g:I\to\mathbb{R}$ be differentiable at a point $x\in I$. Then:

• (Linearity) For constants $c\in\mathbb{R}$,

  $$(f+g)'(x)=f'(x)+g'(x),\qquad (cf)'(x)=c\,f'(x).$$

• (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 for the composition $f\circ g$,

  $$(f\circ g)'(x)=f'(g(x))\,g'(x).$$

These identities are the basic computational tools for the derivative and are organized and extended in the chain rule and related results (for example, rules used in local inversion ).