Chain Rule - The Chain Rule in Higher Dimensions

The Chain Rule in Higher Dimensions

The simplest generalization of the chain rule to higher dimensions uses the total derivative. The total derivative is a linear transformation that captures how the function changes in all directions. Let f : Rm → Rk and g : Rn → Rm be differentiable functions, and let D be the total derivative operator. If a is a point in Rn, then the higher dimensional chain rule says that:

or for short,

In terms of Jacobian matrices, the rule says

That is, the Jacobian of the composite function is the product of the Jacobians of the composed functions. The higher-dimensional chain rule can be proved using a technique similar to the second proof given above.

The higher-dimensional chain rule is a generalization of the one-dimensional chain rule. If k, m, and n are 1, so that f : R → R and g : R → R, then the Jacobian matrices of f and g are 1 × 1. Specifically, they are:

The Jacobian of f ∘ g is the product of these 1 × 1 matrices, so it is f′(g(a))g′(a), as expected from the one-dimensional chain rule. In the language of linear transformations, D_a(g) is the function which scales a vector by a factor of g′(a) and D_g(a)(f) is the function which scales a vector by a factor of f′(g(a)). The chain rule says that the composite of these two linear transformations is the linear transformation D_a(f ∘ g), and therefore it is the function that scales a vector by f′(g(a))g′(a).

Another way of writing the chain rule is used when f and g are expressed in terms of their components as y = f(u) = (f₁(u), ..., f_k(u)) and u = g(x) = (g₁(x), ..., g_m(x)). In this case, the above rule for Jacobian matrices is usually written as:

The chain rule for total derivatives implies a chain rule for partial derivatives. Recall that when the total derivative exists, the partial derivative in the ith coordinate direction is found by multiplying the Jacobian matrix by the ith basis vector. By doing this to the formula above, we find:

Since the entries of the Jacobian matrix are partial derivatives, we may simplify the above formula to get:

More conceptually, this rule expresses the fact that a change in the x_i direction may change all of g₁ through g_k, and any of these changes may affect f.

In the special case where k = 1, so that f is a real-valued function, then this formula simplifies even further: