multivariable differential for inner product(scalar product)?

In calculus of a single variable, it is well know that the derivative of product $\displaystyle fg$ is $\displaystyle f'g+fg'$. This formula holds also when the function is vector-valued of a single variable, or complex-valued of a complex variable. But if $\displaystyle \mathbf{f,g}$ are general functions from $\displaystyle {\mathbb R}^n$ into $\displaystyle {\mathbb R}^m$, do we have a similar formula for the differential(total derivative) of their inner product(scalar product) $\displaystyle \mathbf{f\cdot g}$ defined as $\displaystyle (\mathbf{f\cdot g)(x)=f(x)\cdot g(x)}$? Thanks.