When you implicitly differentiate , you are differentiating with respect to a particular variable—in this case, x, so:
From the 3rd line to the 4th line, is the derivative with respect to x of , in which (as in Ryan Budney's comment) we assume that y is some function of x, so we apply the chain rule, differentiating with respect to y and multiplying by the derivative of y with respect to x to get .
edit: I think it might be useful if I introduced a slightly different notation: Let be the differential operator with respect to x, which you have previously written as (and, similarly, is the differential operator with respect to y). When we apply the differential operator to something, we read and write it like a function: is "the derivative with respect to x of is ."
Now, rewriting the work above in this notation:
And, to your question of finding :
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