Product rule for differentiation of dot product

Statement

Statement for vector-valued functions of a single variable

Version type	Statement
Specific point, named functions	Suppose $f$ and $g$ are $n$-dimensional. vector-valued functions of a single variable and $x_{0}$ is a point in the domain such that $\!f'(x_{0})$ and $\!g'(x_{0})$ both exist (again, as $n$-dimensional vectors). Then $(f\cdot g)'(x_{0})=f'(x_{0})\cdot g(x_{0})+f(x_{0})\cdot g'(x_{0})$ where $\cdot$ denotes the dot product.