Using the alternative definition for a derivative, the limit as Z approaches X of F of Z -, F of X / Z -, X to find the derivative of the following function. So the first thing we really need to know is what is F of Z? And we're going to stick Z in every time we see the X. So we get F of Z equaling 1 + sqrt Z. So using this definition limit Z approaches X of the quantity one plus square to Z minus the quantity 1 + sqrt X all divided by Z -, X. When we distribute out the -1, the ones are going to cancel, so we end up with just sqrt Z -, sqrt X. Now to get rid of that, we need to multiply by its conjugate. And if we're going to do the top, we have to also do the bottom. The reason we have to be doing this is we need to figure out how to get rid of the C -, X in the bottom because that C -, X would give us zero if we stuck Z approaches X the way it is right now. So we're going to foil at the top square to Z ^2 plus square to ZX minus square to ZX minus sqrt X ^2. The square roots and the squares cancel, so we get Z. The inner two terms are going to cancel because that's what a conjugate should be doing, and the square root and the square at the end going to cancel. So now we have the Z -, X / Z -, X which is going to reduce to one, and we get 1 divided by square to Z + sqrt X. Now as Z goes to X, we just stick X and when we see our Z so we get 1 / sqrt X + sqrt X 1 / 2 square roots of X.