I am trying to take the derivative of an inner product (in the most general sense over L^2), and was curious if the derivative follows the "chain rule" for inner products. i.e. Does D_y(<f,g>) = <D_y(f),g> + <f,D_y(g)> where D_y is the partial derivative w.r.t. y. So for example, IT IS TRUE that if f=x*y and g=sin(x*y) and the inner product <f,g> = Integral(f#g, w.r.t. x,-Pi,+Pi), f# = complex conj. of f. then the equality holds. In other words, differentiating w.r.t. y and integrating w.r.t x the forumla holds. It seems more trivial if the variable which is being differentiated & integrated is the same. But is it true in general? What if we are differentiating more abstract inner products (i.e. not necessarily integration).