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