I am not exactly sure what you are asking but as

**Aryth** said it seems to use the

implicit function theorem. In its most basic form it says that if

(this notation it means that

is a function of two variables) is a

function (ignore what this mean - it just means the function is behaved well-enough such as being differenciable). Say that

. The implicit function theorem says that if

then the equation

can be solved

__uniquely__ (for

) in the "neighborhood" of

. The term "in the neighborhood" means in some small disk around

we can solve this equation uniquely for

(in terms of

).

For example, consider the circle

. Where can we solve for

uniquely (in a neighborhood)? Define the function

. The point

lies on this circle because

. If we try to solve the equation within the small disk

we see that it has two solutions, one in the upper half and one in the lower half (see red circle) - in fact those solutions are

. And this happens no matter how small we make the circle. Therefore, the equation

**cannot** be solved

__uniquely__. Let us see what happens when we use the implicit function theorem. The theorem says that if

then we can solve the equation uniquely. Since we cannot solve this equation uniquely it must mean that

. Check this:

and so

. However, if

and

i.e. we stay away from the break in the graph where it lies above and below the x-axis then

and therefore we can solve this equation

__uniquely__. As in the case of the blue circle, the solution is

.

Let us say that

is function which satisfies the condition of the implicit function theorem as in the first paragraph. Then in the neighborhood of

(where

is a solution to

) we can solve for

__uniquely__ in terms of

. This means we can define a new function

in this neighborhood. Remember a function is a set of pairs so that for each first coordinate there is a

__unique__ matching coordinate. Therefore, if we let

be the first coordinate then by the theorem there is a

__unique__ matching coordinate so that

. And this would define a function

.

But the implicit function theorem does not stop here, it says more it says that

is itself

(which is differenciable with some more properties). Now since

. The functions

and

are differenciable and so by the

chain rule for multivariable functions we get that (differenciating both sides of

)

and this means

(the partials are evaluated at

). And that gives you a formula.

If you got that then great. There is just one more point to be addressed. If it confuses you just ignore that. The questionable step is dividing by

. How do we know it is non-zero? This is where we use two facts. The first one is that

. The second one is that

is

. The first fact is not good enough to say that

for all points around

because it might be non-zero at

and yet be zero at some points around

. This is where we need the second fact. The meaning of

is that the function is differenciable

**and** the derivative is continous. Therefore

is a continous function. Since

it means there is a small enough neighborhood so that

for points close to

by the definition of continuity.