Your question is a little vague, but I'll try.<p>What it is: Differentiate both sides of an equation. That's it.<p>Why it works: When two things are equal, if we transform them in identical ways (being careful about what "identical" means), then the results are equal. So if two differentiable functions are equal, then their derivatives must be equal.<p>What it's good for: In spite of all the carefully selected examples we're given in calculus class, the fact is that solving equations is often very difficult, and usually <i>impossible</i>. So if someone writes some "f(x,y) = k", and asks you to solve for y in terms of x, well, in general, you can't do it.<p>Now, suppose you want to find dy/dx in the above situation. The "usual" way is to solve for y and then differentiate. But what if you cannot do that? If you can find the derivative, w.r.t. x, of f(x,y), then you can apply implicit differentiation, to get something like g(x,y)*dy/dx + h(x,y) = 0. Then finding dy/dx is easy.<p>So: What it is good for is finding a derivative while avoiding the pain (or the impossibility) of solving an equation.<p>Of course, that leads to the question of what derivatives are good for, but I'm hoping you have some inkling of the answer to that one. If not, then back up, and learn more about derivatives.