Given a function of two variables from $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$, we write/denote this function as $$f:(x,y)\mapsto (u(x,y), v(x,y)).$$ Consider another function, the polar coordinate function $g:(r,\theta)\mapsto (r\cos\theta, r\sin\theta).$ Writing components as functions, $$g:(r,\theta)\mapsto (a(r,\theta),b(r,\theta)).$$ The composition is given by $$(r,\theta)\mapsto (u(a(r,\theta), b(r,\theta)), v(a(r,\theta), b(r,\theta))).$$ This looks (at least initially) too complicated.

Now in the case of functions of single variables, writing and deriving the chain rule of derivative of composite is not too difficult - it comes from definition; but I was in trouble to proceed in functions of two variables.

When explaining in the class, the class immediately became in sleepy mode when I went to derive formula for partial derivatives of composition. This bothered me, isn't there an easy transparent approach to derive this? Isn't it possible to ask students themselves to solve certain small steps and arrive at the formula for partial derivatives of composite?

Question: What is the simple way to derive formula for partial derivatives of the composition from definition of derivatives?

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    $\begingroup$ The simple way IMO is to compute the differential of the composite function, which is a matter of a few matrix multiplications. Specific partial derivatives can then be pulled out of the result. $\endgroup$ – amd May 8 at 4:04

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