Let $f(x) = 3 \sqrt{x} + 1$. If $h \neq 0$, then the difference quotient can be simplified as $$\frac{f(x + h) - f(x)}{h} = \frac{A}{\sqrt{Bx + Ch} + \sqrt{x} },$$ where $A$, $B$, and $C$ are constants. (Note: It's possible for one or more of these constants to be 0.) Find the constants.

$A =$ , $B =$ , and $C =$ .

Use your answer from above to find $\displaystyle{f'(x) = \lim_{h\rightarrow 0} \frac{f(x + h) - f(x)}{h}} =$ .

Finally, find each of the following:

$f'(1) =$ , $f'(2) =$ , and $f'(3) =$ .