Define by where and .
Now we will define a new function (of two variable) in the following way: where are the unique points such that . In other words, we define to be evaluated at a point to be evaluated at where is the (unique point in ) that is mapped into under the transformation that you gave.
It should be clear that i.e. (function composition) for each point .
Now we are going to use the chain rule. That is where I am going to be a little informal with notation because it gets cleaner to write this way and it perhaps be easier for you to follow.
Let us analyze what the first equation means (the second one is similar). It is saying that (by chain rule) we get the expression on the right hand side. When we write we mean the partial derivative with respect to the first coordinate evaluated at (that is at ) multiplied by - the partial derivative of the function with respect to the first coordinate. Similarly, means the partial derivative with respect to the second coordinate evaluated at (that is ).
Therefore, we get upon writing everything out,
Can you finish now comrade?