Proof of a chain rule for partial derivatives

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Simple proof of a basic chain rule for partial derivatives. The notion of differentiability is incorporated into the proof. Such ideas have lots of applications in university mathematics.

[@nicholascousar4306]

Would you do a proof of the chain rule involving two intermediary variables each depending on two independent variables? For instance, let w be a differentiable function of x and y, and let x and y be differentiable functions of r and s. The only explanations I can find involve tree diagrams, and that seems to hand-wavey.

[@DrChrisTisdell]

Thanks. Glad you found it helpful.

[@lcmonon]

Thanks!!!!! This was quite helpful

[@zh1412]

Thanks for the video explanation!!

[@someonetoogoodforyou]

awesome sauce

[@the_sophile]

thanks

[@user-oz8px2ji6r]

How to differentiate w with respect to x? x=g(r,s)

[@Leo-io4bq]

You didn't define the domain and co-domain. Could you please do that

[@DrChrisTisdell]

An example would be R and R^2 👍

[@Leo-io4bq]

Wow you really replied after 11 years.
So, since R^2 = C this proof also holds for komplex functions, right?