How do I do implicit differentiation?

The main thing to remember with implicit differentiation is to differentiate each thing as you see it. Don't worry about rearranging the equation until you have differentiated everything first. Say you have the question,

Differentiate x^2 + y^2 = 1 with respect to x.

Starting from the left, the differential of x^2 with respect to x is 2x. 

Next we have y^2. When we have a term that is not what we are differentiating with respect to (ie y is not x) then you differentiate the term as you would do normally but you have to put a dy/dx on the end. So the differential of y^2 with respect to x is 2y(dy/dx).

And of course the differential of 1 with respect to 1 is 0 (as is the case with any constant.)

This leaves us with a result of:

2x + 2y(dy/dx) = 0

Often this can be your final answer but if you are asked to make dy/dx the subject then you just rearrange the equation to make this the case.

If you do this correctly, you will find your answer to be dy/dx = - (x/y)