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Use the definition of the derivative to find $f'(x)$ if $f(x) = \frac{3}{x^{0.5}+2} , x>0$

To begin with the definition is $$f'(x) = \lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h}$$

Thus, this is what I have so far,

$$f'(x) = \lim_{h\rightarrow0} \frac{f(x+h)-f(x)}{h}= \lim_{h\rightarrow0} \frac{\frac{3}{(x+h)^{0.5}+2} - \frac{3}{(x^{0.5})+2)}}{h}$$

I dont know how to simplify this.. But using the quotient rule, I found that the derivative is $$\frac{-3}{2x^{0.5}(x^{0.5}+x)^2}$$

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$h\rightarrow 0$ not $x\rightarrow 0$.

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  • $\begingroup$ Oh yes, thank you. $\endgroup$ – stacks-geek Oct 10 '14 at 0:30
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Try finding a common denominator first. Then use a conjugate.

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