<s>[INST] How to find the derivative of $f(x)$? This is not a homework problem.
How to find $f'(x)$ if $f(x)=2x^{1/2}\log(2)$?
Thanks in advance for your help! I just can't figure it out...
What rule should I use?
 [/INST] Note that $\frac{d}{dx} (c \cdot f(x)) = c \cdot f'(x)$. Hence you just want to find the derivative of $f(x) = \sqrt{x}$.  Using the definition of the derivative we have 
\begin{align*}
f'(x) & = \lim_{h \to 0} \: \frac{f(x+h)- f(x)}{h} \\ &= \lim_{h \to 0} \: \frac{\sqrt{x+h} - \sqrt{x}}{h} \\ &= \lim_{h \to 0} \: \frac{\sqrt{x+h}-\sqrt{x}}{h} \times \frac{\sqrt{x+h} + \sqrt{x}}{\sqrt{x+h}+\sqrt{x}} \\ &=\lim_{h \to 0} \: \frac{x+h - x}{h \cdot \bigl(\sqrt{x+h} + \sqrt{x}\:\bigr)} \\ &= \lim_{h \to 0} \frac{1}{\sqrt{x+h} + \sqrt{x}} = \frac{1}{2 \cdot \sqrt{x}}
\end{align*}
Use the above rule and see that $ \displaystyle f'(x) = 2 \cdot \log(2) \cdot \frac{1}{2 \cdot \sqrt{x}}$
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