Using the definition, find the derivative of \( f(x)=\sqrt{x} \) at \( x=1 \)

\( f^\prime (1) = \lim \limits _{h \rightarrow 0} \frac{\sqrt{1+h}-\sqrt{1}}{h} \) \( =\lim \limits _{h \rightarrow 0} \frac{\sqrt{1+h}-1}{h} \) \( =\lim \limits _{h \rightarrow 0} \frac{\sqrt{1+h}-1}{h}\frac{\sqrt{1+h}+1}{\sqrt{1+h}+1} \) \( =\lim \limits _{h \rightarrow 0} \frac{1+h-1}{h(\sqrt{1+h}+1)} \) \( =\lim \limits _{h \rightarrow 0} \frac{1}{\sqrt{1+h}+1}=\frac{1}{\sqrt{1+0}+1}=\frac{1}{2} \)