If (V,⟨⋅,⋅⟩) is a finite-dimensional inner product space and f,g:R⟶V are differentiable functions, a straightforward calculation with components shows that
This approach is not very satisfying. However, attempting to apply the definition of the derivative directly doesn’t seem to work for me. Is there a slick, perhaps intrinsic way, to prove this that doesn’t involve working in coordinates?
As h→0 the first expression converges to
ddt⟨f(t),g(t)⟩ and the last expression converges to
by definition of the derivative, by continuity of g and by continuity of the scalar product. Hence the desired equality follows.
Note that this doesn’t use finite-dimensionality and that the argument is the exact same as the one for the ordinary product rule from calculus.