<s>[INST] Use limit definition to find derivative of $x+\sqrt x$ The function is $f(x) = x + \sqrt x$.
How would you use the limit definition of the derivative to find the derivative of that equation?
 [/INST] We have 

$$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$

Let $$f(x) = x + \sqrt x$$
$$\begin{align}f'(x)&=\lim_{h\to0}\frac{(x+h + \sqrt {x+h})-(x + \sqrt x)}{h}\\
&=\lim_{h\to0}\frac{h + \sqrt {x+h} - \sqrt x}{h}\\
&=\lim_{h\to0}1+\frac{ \sqrt {x+h} - \sqrt x}{h}\\
&=\lim_{h\to0}1+\frac{ \sqrt {x+h} - \sqrt x}{h}\frac{\sqrt {x+h} + \sqrt x}{\sqrt {x+h} + \sqrt x}\\
&=\lim_{h\to0}1+\frac{x+h - x}{h( \sqrt {x+h} + \sqrt x)}\\
&=\lim_{h\to0}1+\frac{h}{h( \sqrt {x+h}+ \sqrt x)}\\
&=\lim_{h\to0}1+\frac{1}{ \sqrt {x+h} + \sqrt x}\\
&=1+\frac{1}{ \sqrt {x} + \sqrt x}\\
f'(x)&=1+\frac{1}{ 2\sqrt {x} }\\
\end{align}$$
</s>