# How should I approach finding the derivative using the limit process, for $f(x) = 8/(x^{1/2})$?

I am setting up the problem to find the limit as delta x goes to zero, using the definition in the correct format. No matter what I've tried, I am stuck without finding the right cancellations to leave me with $-4/x^{1.5}$

Would you help me get unstuck?

I have tried everything I can think of and I am missing some little algebra technique. Any help would be much appreciated.

Added Note: A word to other calculus and precalculus beginners - don't forget how to tie your shoes just because you're working with new ideas like Limits and derivatives. Algebra still works, but you will encounter new opportunities for ingenuity not usually seen in most precalculus or algebra classes.

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## 1 Answer

$$\frac1h\left(\frac8{\sqrt{x+h}}-\frac8{\sqrt{x}}\right)=\frac{-8}{\sqrt{x}\sqrt{x+h}(\sqrt{x}+\sqrt{x+h})}.$$ Hint: $\sqrt{a}-\sqrt{b}=\frac{a-b}{\sqrt{a}+\sqrt{b}}$.

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Thanks for trying to help me. Where I am stuck is when I work with the left hand expression, I multiplied through by the product of the denominators to create a common denominator. What I am left with still gives me 0/0. I do not understand how you got your right hand side, nor what to do with it to arrive at my known answer. Do you see where I got stuck? A gentle nudge would be appreciated. –  FreeTrader Oct 10 '11 at 7:55
Once you write the difference as a single fraction, there sould be a term of the form $\sqrt{a}-\sqrt{b}$ in the numerator. Nudge: use the hint on this term. –  Did Oct 10 '11 at 7:59
What is the function you are trying to find the limit for? –  Emmad Kareem Oct 10 '11 at 9:21
Emmad, it is 8 / (x^1/2) –  FreeTrader Oct 10 '11 at 9:28
Yes, I understand that Didier. I think I should not have taken a shortcut when answering to Emmad. your hint above is a very good hint Didier. Is this term a regular feature when multiplying by a conjugate? –  FreeTrader Oct 10 '11 at 10:06
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