derivada
02 nov 2013, 01:10
determine a derivada.
\(f(x)=\sqrt[3]{\frac{x-1}{x+1}}\)
\(f(x)=\sqrt[3]{\frac{x-1}{x+1}}\)
Re: derivada [resolvida]
02 nov 2013, 01:26
olá 
\(\text{Uma otima tecnica para derivar e inserir ln (logaritmo natural dos dois lados) :\)
\(\text{Sabemos que y=\sqrt[3]{\frac{x-1}{x+1}}\)
\(\text{ficando com:}\)
\(\\\\ lny=ln(\sqrt[3]{\frac{x-1}{x+1}})\)
\(\\\\ lny=\frac{1}{3}*(ln({\frac{x-1}{x+1}))\)
\(\\\\ lny=\frac{1}{3}*(ln(x-1)-ln(x+1))\)
\(\text{Derivando:}\)
\(\\\\ \frac{y'}{y}=\frac{1}{3}*(\frac{1}{x-1}-\frac{1}{x+1})\)
\(\\\\ y'=y*(\frac{1}{3}*(\frac{1}{x-1}-\frac{1}{x+1}))\)
\(\\\\ y'=\sqrt[3]{\frac{x-1}{x+1}}*(\frac{1}{3}*(\frac{1}{x-1}-\frac{1}{x+1}))\)
\(\\\\ y'=\sqrt[3]{\frac{x-1}{x+1}}*(\frac{1}{3}*(\frac{2}{x^{2}-1}))\)
\(\text{Uma otima tecnica para derivar e inserir ln (logaritmo natural dos dois lados) :\)
\(\text{Sabemos que y=\sqrt[3]{\frac{x-1}{x+1}}\)
\(\text{ficando com:}\)
\(\\\\ lny=ln(\sqrt[3]{\frac{x-1}{x+1}})\)
\(\\\\ lny=\frac{1}{3}*(ln({\frac{x-1}{x+1}))\)
\(\\\\ lny=\frac{1}{3}*(ln(x-1)-ln(x+1))\)
\(\text{Derivando:}\)
\(\\\\ \frac{y'}{y}=\frac{1}{3}*(\frac{1}{x-1}-\frac{1}{x+1})\)
\(\\\\ y'=y*(\frac{1}{3}*(\frac{1}{x-1}-\frac{1}{x+1}))\)
\(\\\\ y'=\sqrt[3]{\frac{x-1}{x+1}}*(\frac{1}{3}*(\frac{1}{x-1}-\frac{1}{x+1}))\)
\(\\\\ y'=\sqrt[3]{\frac{x-1}{x+1}}*(\frac{1}{3}*(\frac{2}{x^{2}-1}))\)