chain rule (multivariable)
If you run two processes in sequence — first stretch a sheet of rubber, then bend it — the total distortion near a point is got by doing one distortion and then the other. The multivariable chain rule says exactly this about derivatives: the linear approximation of a composition is the composition of the linear approximations. In matrix language, you multiply the Jacobians.
Precisely, if g: R^n -> R^m is differentiable at a and f: R^m -> R^k is differentiable at b = g(a), then the composition f o g is differentiable at a, and its total derivative is the composite linear map D(f o g)(a) = Df(b) Dg(a). In matrices this is Jacobian multiplication: the Jacobian of the composite is the product of the two Jacobians, in that order.
All the messy partial-derivative chain rules of multivariable calculus — sums over intermediate variables, df/dt = sum (df/dx_i)(dx_i/dt) along a path — are just the entries of this one matrix product written out. The clean linear-map statement is both easier to remember and easier to prove, and it makes the order of multiplication (which matters, since matrices do not commute) unambiguous.
Let z = f(x, y) and x = x(t), y = y(t). Then dz/dt = (df/dx)(dx/dt) + (df/dy)(dy/dt). This is the row vector grad f times the column vector (dx/dt, dy/dt) — a 1-by-2 times 2-by-1 Jacobian product.
The single-path chain rule is one entry of the matrix product.