(I use the fx notation style here 'cause it's easier than ∂f/∂x.) My calculus book takes it as a given--or gives it as a given, rather--without any sort of proof even in the appendix.The limit definition of a derivative, has, however, provided me with almost a proof: fxy = limΔy→0( (1/Δy)[fx(x,y+Δy,z) - fx(x,y,z)] ) = limΔy→0{ (1/Δy)[ limΔx→0( (1/Δx)[ f(x+Δx,y+Δy,z) - f(x,y+Δy,z) ] ) - limΔx→0…