Definition · applies to functions of several variables · also called differential, Fréchet derivative, $Df(x)$

Undergraduate

In multivariable calculus, the total derivative of $f : \reals^n \to \reals^m$ at $x$ is the linear map $Df(x)$ that best approximates $f(x + h) - f(x)$ for small $h$.

Examples

Worked examples

  • $f(t, x) = t x^2$ : The linear map $Df(t, x) : (\Delta t, \Delta x) \mapsto x^2 \Delta t + 2 t x \Delta x$ Undergraduate

Notes

The matrix of a total derivative is the Jacobian matrix.

This is the same concept as the differential. See the meanings of “total derivative”.

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