# Differentiable function/D in R/Rules/Fact

Let ${\displaystyle {}D\subseteq \mathbb {R} }$ be a subset, ${\displaystyle {}a\in D}$ a point, and

${\displaystyle f,g\colon D\longrightarrow \mathbb {R} }$

functions which are differentiable in ${\displaystyle {}a}$. Then the following rules for differentiability holds.

1. The sum ${\displaystyle {}f+g}$ is differentiable in ${\displaystyle {}a}$, with
${\displaystyle {}(f+g)'(a)=f'(a)+g'(a)\,.}$
2. The product ${\displaystyle {}f\cdot g}$ is differentiable in ${\displaystyle {}a}$, with
${\displaystyle {}(f\cdot g)'(a)=f'(a)g(a)+f(a)g'(a)\,.}$
3. For ${\displaystyle {}c\in \mathbb {R} }$, also ${\displaystyle {}cf}$ is differentiable in ${\displaystyle {}a}$, with
${\displaystyle {}(cf)'(a)=cf'(a)\,.}$
4. If ${\displaystyle {}g}$ has no zero in ${\displaystyle {}a}$, then ${\displaystyle {}1/g}$ is differentiable in ${\displaystyle {}a}$, with
${\displaystyle {}{\left({\frac {1}{g}}\right)}'(a)={\frac {-g'(a)}{(g(a))^{2}}}\,.}$
5. If ${\displaystyle {}g}$ has no zero in ${\displaystyle {}a}$, then ${\displaystyle {}f/g}$ is differentiable in ${\displaystyle {}a}$, with
${\displaystyle {}{\left({\frac {f}{g}}\right)}'(a)={\frac {f'(a)g(a)-f(a)g'(a)}{(g(a))^{2}}}\,.}$