Total derivative

Summary

Let $f$ map an open subset of $\mathbb{R}^n$ to $\mathbb{R}^m$. The total derivative of $f$ at a point $a$ is the linear map $Df(a) \colon \mathbb{R}^n \to \mathbb{R}^m$ that best approximates $f$ near $a$. That is, $f(a + h) = f(a) + Df(a)(h) + o(\|h\|)$ as $h \to 0$. It is the Fréchet derivative in the case of Euclidean spaces. Its matrix in the standard bases is the Jacobian matrix of $f$ at $a$.

Remarks

The total derivative accounts for changes in all variables at once. A partial derivative accounts for a change in one variable only. When $f$ depends on $t$ both directly and through other functions of $t$, the total derivative $\dfrac{d}{dt} f(t, x(t))$ is the sum $\dfrac{\partial f}{\partial t} + \dfrac{\partial f}{\partial x} \dfrac{dx}{dt}$, by the chain rule.

Explanations

L1 human

For a function of one variable, the derivative at a point is a number: the slope of the tangent line. For a function of several variables, the derivative at a point is a linear map: it tells how the output changes for each small step in any direction.