Differentiable function/D open in K/Rules/Fact

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

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

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

1. The sum ${}f+g$ is differentiable in ${}a$ with

   ${}(f+g)'(a)=f'(a)+g'(a)\,.$

2. The product ${}f\cdot g$ is differentiable in ${}a$ with

   ${}(f\cdot g)'(a)=f'(a)g(a)+f(a)g'(a)\,.$

3. For ${}c\in {\mathbb {K} }$ also ${}cf$ in ${}a$ differentiable with

   ${}(cf)'(a)=cf'(a)\,.$

4. If ${}g$ has no zero in ${}a$, then ${}1/g$ is differentiable in ${}a$ with

   ${}{\left({\frac {1}{g}}\right)}'(a)={\frac {-g'(a)}{(g(a))^{2}}}\,.$

5. If ${}g$ has no zero in ${}a$, then ${}f/g$ id differentiable in ${}a$ with

   ${}{\left({\frac {f}{g}}\right)}'(a)={\frac {f'(a)g(a)-f(a)g'(a)}{(g(a))^{2}}}\,.$

Proof, Write another proof