How do I implicitly differentiate and why does it work? (Assuming understanding of differentiation)

Implicit differentiation can be used when you are asked to find dy/dx of a function that has not been written as y=f(x) e.g. y = x^2 - 1, and which cannot be rearranged as such. We can use the equation of a circle as an example, x^2 +y^2 = 25. In order to implicitly differentiate we have to differentiate each term with respect to x, this is straight forward for the x^2 and 25 terms but for any term which is a function of y we differentiate pretending that y is just another x term and then multiply that by dy/dx. e.g. y^2 -> 2ydy/dx. Once all the terms have been dealt with we can rearrange to find dy/dx.Why does this work? Let's consider what differentiating a function of y with respect to y looks like: df(y)/dy, but we need to find df(y)/dx so if we times df(y)/dy x dy/dx we can see that the product is now df(y)/dx for that term.