Differentiating an Inner Product

If (V,,) is a finite-dimensional inner product space and f,g:RV 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?


Observe that
As h0 the first expression converges to
ddtf(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.

Source : Link , Question Author : ItsNotObvious , Answer Author : t.b.

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