(adsbygoogle = window.adsbygoogle || []).push({}); Is there such a thing as a total "partial" derivative?

Total Derivative as I've Been Taught

From my understanding, if we have a function s = f(x, y) where the two arguments x and y are related by another function y = g(x), then there is a great deal of difference between ds/dx and ∂s/∂x.

∂s/∂x is simply a partial derivative and can be calculated by treating y as a constant and differentiating f(x, y) with respect to x.

On the other hand, the "total derivative" ds/dx takes the y = g(x) relationship into account and, by the Chain Rule, gives:

[itex]\frac{ds}{dx}[/itex] = [itex]\frac{∂s}{∂x}[/itex][itex]\frac{dx}{dx}[/itex] + [itex]\frac{∂s}{∂y}[/itex][itex]\frac{dy}{dx}[/itex]

This approach is very well explained in Wikipedia:

http://en.wikipedia.org/wiki/Total_derivative#Differentiation_with_indirect_dependencies

A Different Case

However, what happens if we have a function s = f(x, y, z) and only two of the arguments are related, as through y = g(x).

As before, ∂s/∂x can still be calculated by differentiating f(x, y, z) and treating y and z as constants, but what of the total derivative in terms of x?

Such a total "partial" derivative would take the form:

(total partial derivative in terms of x) = [itex]\frac{∂s}{∂x}[/itex][itex]\frac{dx}{dx}[/itex] + [itex]\frac{∂s}{∂y}[/itex][itex]\frac{dy}{dx}[/itex]

But, clearly, we can't notate this as ds/dx since s is also a function of z. Neither can we call is ∂s/∂x since that notation is reserved for the regular partial derivative.

So my question: is there such a concept as a total "partial" derivative"? I haven't been able to find any discussion on such a concept and was curious about whether something like this even exists.

Any replies are appreciated, and thank you in advance!

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# Is there such a thing as a total partial derivative?

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