Irrational denominator limit derivative

[Orion1]

Can anyone provide some information for this formula?

I tried rationalizing the denominator and cross multiplication and combining terms, and also multiplying by the conjugate of the demoninator, what am I doing wrong?

$$f(x) = \frac{1}{\sqrt{x + 2}} \; \; \; \text{find} \; f'(a)$$

$$f'(a) = \lim_{h \rightarrow 0} (\frac{1}{\sqrt{(a + h) + 2}} - \frac{1}{\sqrt{a + 2}}) \frac{1}{h}$$

 

[Robokapp]

Ok. don't write it any more complicated than it already is. It's actaully not very hard...

write it as $$f(x)=(x+2)^{-1/2}$$ and apply the chain rule. Or power Rule...I don't know them by name. Especially since x has no coefficient it's very "clean".

to answer the "what I'm doing wrong" part...you don't want to apply the $$\frac{f(x+h)-f(x)} {h}$$ unless absolutely everything else fails. I mean once you're past chapter 2 calculus AB it becomes unused until maybe way later which I'd not know.

 

[HallsofIvy]

Robokapp, you should consider the possibility that Orion1 is required to use the basic definition of the derivative for practice. Of course, if that's the case, then this is school work so I am moving this thread to the homework section!

Orion1, first subtract the two fractions, getting
$$\frac{\sqrt{x+2}- \sqrt{x+h+2}}{\sqrt{x+2}\sqrt{x+h+2}}$$
then try "rationalizing the numerator": multiply both numerator and denominator by
$$\sqrt{x+2}+ \sqrt{x+h+2}$$