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* × *Y* → *Z* 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* × *Y* → *Z* defined by

D_{(x,y)}*B*(*u,v*) = *B*(*x,v*) + *B*(*u,y*)
for every (

*u,v*)∈

*X×Y*.

*This writeup is in the public domain.*