Total Derivative

The derivative as the linear map that best approximates a function.

The idea

The total derivative carries differentiation to functions of several variables, and the reading of $f'(a)$ that generalizes is not the number but the linear approximation: $f'(a)$ names the linear function $h \mapsto f'(a)h$ whose graph is the tangent line, the best straight-line approximation to $f$ near $a$.

Let $f \colon \mathbb{R}^{n} \to \mathbb{R}^{m}$ and let $a$ be an interior point of its domain. We call $f$ differentiable at $a$ if there is a linear map $L \colon \mathbb{R}^{n} \to \mathbb{R}^{m}$ with $\lim_{h \to 0} \frac{\lVert f(a+h) - f(a) - Lh \rVert}{\lVert h \rVert} = 0,$ equivalently $f(a+h) = f(a) + Lh + o(\lVert h \rVert)$: the error of the approximation $f(a) + Lh$ must vanish faster than $\lVert h \rVert$ itself, not merely vanish. At most one linear map satisfies this, and it is the total derivative $Df(a)$.

A derivative in several variables is therefore a linear map, not a number and not a list of numbers. Its matrix in the standard bases has the partial derivatives as entries, which is how we compute it, but the entries are not the definition: a function can have every partial derivative at $a$ and still fail to be differentiable there, because the partials probe $f$ only along the coordinate axes, while the limit above must hold as $h$ approaches $0$ from every direction at once.

Ways to work on it

• Walkthrough. The defining property of the total derivative, its matrix of partials, and linear approximation.
• Practice. Evaluate an entry of a quadratic field's derivative matrix.
• Hardest. A Jacobian entry, and why partials can exist without differentiability.

Not sure where to start? Take the ten-question placement test.