If I was trying to prove the chain rule for partial derivatives, can I start with the definition of a total differential? What I mean is:(adsbygoogle = window.adsbygoogle || []).push({});

Let [tex]f(x,y)=z[/tex] where [tex]x=g(t)[/tex] and [tex]y=h(t)[/tex].

I'm looking for [tex]\frac{dz}{dt}[/tex].

By definition,

[tex]dz = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y}dy[/tex]

So, can I simply divide both sides by [tex]dt[/tex] and obtain,

[tex]\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}[/tex]

Which happens to be the partial derivate using chain rule, and thus proving it? Or must I go with the definitions of the derivatives ([tex]\Delta[/tex] and all)?

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# Proving chain rule

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