Example · Undergraduate
Terms
| Term | This example | Why |
|---|
| partial derivative |
$\frac{\partial f}{\partial t} = x^2$ and $\frac{\partial f}{\partial x} = 2 t x$
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$f$ has two variables, so it has two partial derivatives, and no ordinary derivative.
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| differential |
The linear map $\mathrm{d}f_{(t, x)} : (\Delta t, \Delta x) \mapsto x^2 \Delta t + 2 t x \Delta x$
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The differential of $f(t, x) = t x^2$ combines both partial derivatives into the linear approximation $Df(t, x)$.
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| total derivative (linear map) |
The linear map $Df(t, x) : (\Delta t, \Delta x) \mapsto x^2 \Delta t + 2 t x \Delta x$
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| ordinary derivative |
None
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The function $f(t, x) = t x^2$ has no ordinary derivative, because it has two variables.
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