Definição:
[tex]f(x) = \sqrt{x}[/tex]
[tex]f'(x) = \displaystyle\lim_{x' \to x} \dfrac{f(x)-f(x')}{x-x'}[/tex]
[tex]f'(x) = \displaystyle\lim_{x' \to x} \dfrac{\sqrt{x}-\sqrt{x'}}{x-x'}[/tex]
[tex]f'(x) = \displaystyle\lim_{x' \to x} \dfrac{\sqrt{x}-\sqrt{x'}}{x-x'} \cdot\left(\dfrac{\sqrt{x}+\sqrt{x'}}{\sqrt{x}+\sqrt{x'}}\right)[/tex]
[tex]f'(x) = \displaystyle\lim_{x' \to x} \dfrac{\sqrt{x}^2-\sqrt{x'}^2}{(x-x')\cdot(\sqrt{x}+\sqrt{x'})}[/tex]
[tex]f'(x) = \displaystyle\lim_{x' \to x} \dfrac{x-x'}{(x-x')\cdot(\sqrt{x}+\sqrt{x'})}[/tex]
[tex]f'(x) = \displaystyle\lim_{x' \to x} \dfrac{1}{(\sqrt{x}+\sqrt{x'})}[/tex]
[tex]\boxed{f'(x) = \dfrac{1}{2\sqrt{x}}}[/tex]
Lei da potência:
[tex]f(x) = a\cdot x^n \implies f'(x) = an\cdot x ^{n-1}[/tex]
Aplicando:
[tex]f(x) = \sqrt{x}[/tex]
[tex]f(x) = x^{\frac{1}{2}}[/tex]
[tex]f'(x) = \dfrac{1}{2} \cdot x^{\frac{1}{2}-1}[/tex]
[tex]f'(x) = \dfrac{1}{2}\cdot x^{-\frac{1}{2}}[/tex]
[tex]\boxed{f'(x) = \dfrac{1}{2\sqrt{x}}}[/tex]
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