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 $f^{\prime }(x_0)$ and $g^{\prime }(x_0)$ both exist (again, as $n$-dimensional vectors). Then $(f\cdot g{)}^{\prime }(x_0)=f^{\prime }(x_0)\cdot g(x_0)+f(x_0)\cdot g^{\prime }(x_0)$ where $\cdot$ denotes the dot product.