Fórum de Matemática
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determine a derivada.


\(f(x)=\sqrt[3]{\frac{x-1}{x+1}}\)

olá


\(\text{Uma otima tecnica para derivar e inserir ln (logaritmo natural dos dois lados) :\)


\(\text{Sabemos que y=\sqrt[3]{\frac{x-1}{x+1}}\)


\(\text{ficando com:}\)


\(\\\\ lny=ln(\sqrt[3]{\frac{x-1}{x+1}})\)

\(\\\\ lny=\frac{1}{3}*(ln({\frac{x-1}{x+1}))\)

\(\\\\ lny=\frac{1}{3}*(ln(x-1)-ln(x+1))\)


\(\text{Derivando:}\)


\(\\\\ \frac{y'}{y}=\frac{1}{3}*(\frac{1}{x-1}-\frac{1}{x+1})\)


\(\\\\ y'=y*(\frac{1}{3}*(\frac{1}{x-1}-\frac{1}{x+1}))\)


\(\\\\ y'=\sqrt[3]{\frac{x-1}{x+1}}*(\frac{1}{3}*(\frac{1}{x-1}-\frac{1}{x+1}))\)

\(\\\\ y'=\sqrt[3]{\frac{x-1}{x+1}}*(\frac{1}{3}*(\frac{2}{x^{2}-1}))\)