### Quiz 3.1:

Question:

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

Solution:

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