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I was looking at the following question:

Differentiating an Inner Product

that was talking about the derivative of an inner product to be:

$$ \frac{d}{dt} \langle f, g \rangle = \langle f(t), g^{\prime}(t) \rangle + \langle f^{\prime}(t), g(t) \rangle $$

However, I was looking for a more general answer when $f$ and $g$ belong to a more general Vector Space. In that case does the equivalent conjecture hold?

$$ \bigtriangledown_w \langle w, x \rangle = \langle w, \bigtriangledown_wx \rangle + \langle \bigtriangledown_ww, x \rangle $$

I believe the above does not hold and that it needs some slight modification to make it work. Does someone know what this modification might be? What is the gradient of a more general inner product.

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  • $\begingroup$ I do not see what is more general in your question compared with the linked one. $\endgroup$
    – daw
    Oct 12, 2014 at 16:08
  • $\begingroup$ taking the gradient and taking the derivative wrt to one variable are different things ... Right? $\endgroup$ Oct 12, 2014 at 17:03
  • $\begingroup$ w might be in a vector space of functions, is what I believe makes my question more general. Intuitively, it has more dimension I guess. $\endgroup$ Oct 12, 2014 at 17:46

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