Our project group is stuck with this question and nobody seems to have clue so far.
Let $f(x)=-8+2\sqrt{x}$. Then the expression $\frac{f(x+h)-f(x)}{h}$ can be written of the form $\frac{A}{(\sqrt{Bx+Ch})+(\sqrt{x})}$, where $A, B,$ and $C$ are constants. (Note: It's possible for one or more of these constants to be $0$. Find the constants. Use your answer from above to find $lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$. Finally, find $f'(1), f'(2),$, and $f'(3)$.
Could somebody help out here?
Thank you.
First of all, plug the function into the definition of the derivative. You should end up with a limit which goes to $\frac{0}{0}$. Now you need to start playing around with the numbers to get it into a more useful form. An important trick for a problem like this is the following: $$\sqrt{a}-\sqrt{b} = (\sqrt{a}-\sqrt{b})*1= (\sqrt{a}-\sqrt{b})*(\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}+\sqrt{b}})$$ Now, by simply multiplying out the above expression, you find that $$\sqrt{a}-\sqrt{b} = \frac{a-b}{\sqrt{a}+\sqrt{b}}$$
Try to use that type of reasoning to get a limit of the form the question asks for.