When studying multivariable calculus, one encounters a whole zoo of product rules: there's one for dot products, there's one for cross products, there's one for matrix products; later in functional analysis one sees yet another product rule for inner products. After a while, one gets the feeling that pretty much every notion of product comes with its own product rule. Isn't there some underlying general rule? Yes there is:

Let X, Y and Z be Banach spaces and let B : X × YZ be a continuous bilinear operator. Then B is differentiable and the derivative at the point (x,y) ∈ X×Y is the linear map D(x,y)B : X × YZ defined by

D(x,y)B(u,v) = B(x,v) + B(u,y)
for every (u,v)∈X×Y.

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