n.xn-1Lista de exercicios resolvidos$a)f\left( x \right)={{x}^{5}}$Resolucao:$f'\left( x \right)=n.{{x}^{n-1}}\Rightarrow f'\left( x \right)=5.{{x}^{5-1}}\Rightarrow f'\left( x \right)=5{{x}^{4}}$$b)f\left( x \right)=\frac{1}{{{x}^{5}}}$Resolucao:$f\left( x \right)=\frac{1}{{{x}^{5}}}\Rightarrow f\left( x \right)={{x}^{-5}}\Rightarrow f'\left( x \right)=-5{{x}^{-5-1}}\Rightarrow $$\Rightarrow f'\left( x \right)=-5{{x}^{-6}}\Rightarrow f'\left( x \right)=\frac{-5}{{{x}^{6}}}$$c)f\left( x \right)=\frac{1}{\sqrt{{{x}^{3}}}}$Resolucao:$f\left( x \right)=\frac{1}{\sqrt{{{x}^{3}}}}\Rightarrow f\left( x \right)={{x}^{\frac{-3}{2}}}\Rightarrow f'\left( x \right)=\frac{-3{{x}^{\frac{-3}{2}-1}}}{2}\Rightarrow $$\Rightarrow f'\left( x \right)=\frac{-3{{x}^{\frac{-5}{2}}}}{2}\Rightarrow f'\left( x \right)=\frac{-3}{2{{x}^{\frac{5}{2}}}}\Rightarrow f'\left( x \right)=\frac{-3}{2\sqrt{{{x}^{5}}}}$ $d)f\left( x \right)=\sqrt[3]{{{x}^{2}}}$Resolucao:$f\left( x \right)=\sqrt[3]{{{x}^{2}}}\Rightarrow f\left( x \right)={{x}^{\frac{2}{3}}}\Rightarrow f'\left( x \right)=\frac{2{{x}^{\frac{2}{3}-1}}}{3}\Rightarrow $$\Rightarrow f'\left( x \right)=\frac{2{{x}^{\frac{-1}{3}}}}{3}\Rightarrow f'\left( x \right)=\frac{2}{3\sqrt[3]{x}}$$e)f\left( x \right)={{x}^{2}}.\sqrt[4]{{{x}^{3}}}$Resolucao:$f\left( x \right)={{x}^{2}}.\sqrt[4]{{{x}^{3}}}\Rightarrow f\left( x \right)={{x}^{2}}.{{x}^{\frac{3}{4}}}\Rightarrow f\left( x \right)={{x}^{\frac{11}{4}}}\Rightarrow$Observacao: $2+\frac{3}{4}=\frac{11}{4}$$\Rightarrow f'\left( x \right)=\frac{11{{x}^{\frac{11}{4}-1}}}{4}\Rightarrow f'\left( x \right)=\frac{11{{x}^{\frac{7}{4}}}}{4}\Rightarrow f'\left( x \right)=\frac{11\sqrt[4]{{{x}^{7}}}}{4}$$f)f\left( x \right)=\frac{\sqrt{{{x}^{5}}}}{\sqrt[5]{{{x}^{2}}}}$Resolucao:$f\left( x \right)=\frac{\sqrt{{{x}^{5}}}}{\sqrt[5]{{{x}^{2}}}}\Rightarrow f\left( x \right)=\frac{{{x}^{\frac{5}{2}}}}{{{x}^{\frac{2}{5}}}}\Rightarrow f\left( x \right)={{x}^{\frac{5}{2}}}.{{x}^{\frac{-2}{5}}}\Rightarrow f\left( x \right)={{x}^{\frac{21}{10}}}\Rightarrow$$\Rightarrow f'\left( x \right)=\frac{21{{x}^{\frac{21}{10}-1}}}{10}\Rightarrow f'\left( x \right)=\frac{21{{x}^{\frac{11}{10}}}}{10}\Rightarrow f'\left( x \right)=\frac{21\sqrt[10]{{{x}^{11}}}}{10}$$g)f\left( x \right)=\frac{{{x}^{3}}}{\sqrt[3]{x}}$Resolucao:$f\left( x \right)=\frac{{{x}^{3}}}{\sqrt[3]{x}}\Rightarrow f\left( x \right)=\frac{{{x}^{3}}}{{{x}^{\frac{1}{3}}}}\Rightarrow f\left( x \right)={{x}^{3}}.{{x}^{\frac{-1}{3}}}\Rightarrow$$\Rightarrow f\left( x \right)={{x}^{\frac{8}{3}}}\Rightarrow f'\left( x \right)=\frac{8{{x}^{\frac{8}{3}-1}}}{3}\Rightarrow f'\left( x \right)=\frac{8{{x}^{\frac{5}{3}}}}{3}\Rightarrow f'\left( x \right)=\frac{8\sqrt[3]{{{x}^{5}}}}{3}$$h)f\left( x \right)=x.\sqrt[5]{{{x}^{3}}}$Resolucao:$f\left( x \right)=x.\sqrt[5]{{{x}^{3}}}\Rightarrow f\left( x \right)={{x}^{1}}.{{x}^{\frac{3}{5}}}\Rightarrow f\left( x \right)={{x}^{\frac{8}{5}}}\Rightarrow$$\Rightarrow f'\left( x \right)=\frac{8{{x}^{\frac{8}{5}-1}}}{5}\Rightarrow f'\left( x \right)=\frac{8{{x}^{\frac{3}{5}}}}{5}\Rightarrow f'\left( x \right)=\frac{8\sqrt[5]{{{x}^{3}}}}{5}$$i)f\left( x \right)=\frac{1}{\sqrt[5]{{{x}^{2}}}}$Resolucao: $f\left( x \right)=\frac{1}{\sqrt[5]{{{x}^{2}}}}\Rightarrow f\left( x \right)={{x}^{\frac{-2}{5}}}\Rightarrow f'\left( x \right)=\frac{-2{{x}^{\frac{-2}{5}-1}}}{5}\Rightarrow $$\Rightarrow f'\left( x \right)=\frac{-2{{x}^{\frac{-7}{5}}}}{5}\Rightarrow f'\left( x \right)=\frac{-2}{5.\sqrt[5]{{{x}^{7}}}}$$j)f\left( x \right)=3x+2$Resolucao:$f\left( x \right)=3x+2\Rightarrow f'\left( x \right)=3.\left( 1 \right).{{x}^{1-1}}+0\Rightarrow f'\left( x \right)=3$Observacao: Derivada de constante e igual a zero.Veja: Tabela de Derivadas ElementaresReferenciasNotas de Calculo Integral Diferencial, Pedro Coelho, Santos, Sao Paulo, 2011. Notas de Calculo Integral Diferencial, Profo Fernando, Santos, Sao Paulo, 2010.