Multivariable Calculus 5 | Total Derivative

Total differentiability in several variables generalizes the familiar “best linear approximation” from single-variable calculus. Instead of approximating a function near a point by a line with slope, multivariable calculus approximates it near a point by a linear map that takes small input changes (vectors) to output changes. The key requirement is that the difference between the function’s actual change and the linear prediction becomes negligible compared with the size of the input change as the input vector approaches the zero vector.

In one dimension, differentiability at a point x₀ means there exists a number B and a remainder term R(h) such that f(x₀+h) = f(x₀) + B·h + R(h), with the remainder shrinking faster than h as h → 0. The multivariable version keeps the same structure but replaces “multiply by a slope” with “apply a linear map.” For a function f: ℝⁿ → ℝᵐ, total differentiability at a point x₀ requires a linear map L (the multivariable derivative) and an error term that becomes small relative to the length of the input perturbation. Concretely, one writes f(x₀+h) = f(x₀) + L(h) + r(h), where the remainder r(h) satisfies r(h)/||h|| → 0 as h → 0. Because h is a vector, the normalization uses the Euclidean (ℓ₂) norm ||h||, i.e., the distance from h to the zero vector. The definition is pointwise: it must hold at the chosen x₀, and it must work along any way h approaches 0.

This linear map L is the “total derivative” and is often denoted Df or DF at the point x₀. When L is represented using a matrix, that matrix is the Jacobian matrix J_F of f at x₀, and the linear approximation becomes ordinary matrix-vector multiplication. The transcript emphasizes that the derivative is not merely a number in higher dimensions—it is an entire linear transformation capturing how all input directions affect all output components.

The discussion also shows how the definition reduces to the one-dimensional case when n = m = 1: the total derivative becomes a 1×1 matrix whose entry is f′(x₀), and L(h) is just f′(x₀)·h. A two-dimensional example then makes the idea concrete. Consider a function that flips two components: f(h₁, h₂) = (h₂, h₁). At the origin, f(0,0) = (0,0), and the function is already linear, so the remainder term is zero. The corresponding Jacobian matrix at (0,0) is [0 1; 1 0], which, when multiplied by the input vector (h₁, h₂), produces (h₂, h₁). That example illustrates how total differentiability can be checked by finding the correct linear map (or Jacobian) that matches the function’s first-order behavior near the point.

The takeaway is that total differentiability is the formal condition guaranteeing a reliable linear approximation in multiple dimensions, with the Jacobian providing the practical matrix form of the derivative.