To find:
The derivative of function f(x) using the first principle.
[tex]f(x)=\sqrt{x}[/tex]Solution:
By the first principle, the derivative of the function f(x) is given by:
[tex]f^{\prime}(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}[/tex]So, the derivative of the given function can be obtained as follows:
[tex]\begin{gathered} f^{\prime}(x)=\lim_{h\to0}\frac{\sqrt{x+h}-\sqrt{x}}{h} \\ =\lim_{h\to0}\frac{\sqrt{x+h}-\sqrt{x}}{h}\times\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}} \\ =\lim_{h\to0}\frac{x+h-x}{h(\sqrt{x+h}+\sqrt{x})} \\ =\lim_{h\to0}\frac{h}{h(\sqrt{x+h}+\sqrt{x})} \\ =\lim_{h\to0}\frac{1}{(\sqrt{x+h}+\sqrt{x})} \\ =\frac{1}{\sqrt{x+0}+\sqrt{x}} \\ =\frac{1}{2\sqrt{x}} \end{gathered}[/tex]Thus, the derivative of the given function is:
[tex]f^{\prime}(x)=\frac{1}{2\sqrt{x}}[/tex]