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I can do basic multivariable calculations using partial and total derivatives. I also know for partial derivatives the existence of all partial derivatives at a point doesn't imply continuity.

Are partial derivatives a special case of the total derivative or just something else entirely? Can someone compare / contrast them?

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    $\begingroup$ Partial derivatives are a part of total derivatives. It's kinda right in the names. ;) $\endgroup$
    – user137731
    Commented May 21, 2015 at 3:45
  • $\begingroup$ Oh, I just thought that was because we only did one "part" at a time lol $\endgroup$ Commented May 21, 2015 at 3:46
  • $\begingroup$ When computing total derivative, you don't assume other variables are constants; for partial derivative you do. $\endgroup$
    – mhp
    Commented May 21, 2015 at 3:49
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    $\begingroup$ A partial derivative is a special case of a directional derivative. A directional derivative exists when the function is differentiable. Differentiability in the general sense is far more subtle than the existence of directional derivatives... see posts like math.stackexchange.com/q/503632/36530 $\endgroup$ Commented May 21, 2015 at 3:50
  • $\begingroup$ @mhp but does the assumption other variables aren't constant mean....that they could be constant...or not constant....hence I asked if partial derivatives were a special case because they assume other variables constant $\endgroup$ Commented May 21, 2015 at 4:00

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Here's something that may help:

If $x$ and $y$ are both independent functions of $t$ and $z=f(x,y)$ then

$$\frac{dz}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}$$

In a sense, to get the total derivative, you add up all of the partial derivatives.

So to answer your question, the partial derivative isn't really a special case of the total derivative; it is, as bye_world pointed out in the comments, a $part$ of the total derivative.

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    $\begingroup$ Your equation as you have it is not correct. $\endgroup$
    – mhp
    Commented May 21, 2015 at 5:19
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The other responses answer your question well, but I would like to add a partial derivative is also a 'total' derivative of some function, which is the one-variable function obtained from your original function by setting all the other variables (different from the one with respect to which you are differentiating) constant. So, if, for concreteness, $f : \mathbb{R}^{2} \rightarrow \mathbb{R}$ and you wish to find the partial derivative w.r.t $x$ at $(x_{0},y_{0})$, then you can define $g(x) = f(x,y_{0})$ and then $g'(x_{0}) = \frac{\partial f}{\partial x} (x_{0},y_{0})$, and the l.h.s is a total derivative, whereas the r.h.s is partial derivative.

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