Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives

Let $\mathbb{R}$ be the real numbers. Let $I\subseteq\mathbb{R}$ be an interval, let $f,g:I\to\mathbb{R}$, let $c\in\mathbb{R}$, and let $x_0\in I$ be an interior point of $I$. Here $f+g$, $cf$ and $fg$ denote the pointwise sum, scalar multiple and product on $I$, given by $(f+g)(z)=f(z)+g(z)$, $(cf)(z)=c\,f(z)$ and $(fg)(z)=f(z)\,g(z)$.

Then the following hold.

1. (Constants) For every $b\in\mathbb{R}$, the function on $I$ with constant value $b$ is differentiable at $x_0$ with derivative $0$.

2. (Sum and constant multiple) If $f$ and $g$ are differentiable at $x_0$, then $f+g$ and $cf$ are differentiable at $x_0$, with

$$(f+g)'(x_0)=f'(x_0)+g'(x_0),\qquad (cf)'(x_0)=c\,f'(x_0).$$

3. (Product rule) If $f$ and $g$ are differentiable at $x_0$, then $fg$ is differentiable at $x_0$, with

$$(fg)'(x_0)=f'(x_0)\,g(x_0)+f(x_0)\,g'(x_0).$$