Question

Termine a derivada da Função $4xy^{3} - x^{2}+ x^{3} -5x+6=0$

Answer 1

$y=\sqrt[3]{\frac{-x^3+x^2+5x-6}{4x}}=(\frac{-x^3+x^2+5x-6}{4x})^{\frac{1}{3}}$

$y'= \frac{1}{3}*(\frac{-x^3+x^2+5x-6}{4x})^{-\frac{2}{3}}*(-3x^2+2x+5)=\frac{1}{3}*\frac{-3x^2+2x+5}{ \sqrt[3]{ (\frac{-x^3+x^2+5x-6}{4x})^2}}$

$\frac{1}{3}*\frac{-3x^2+2x+5}{ \frac{\sqrt[3]{(-x^3+x^2+5x-6)^2}}{\sqrt[3]{(4x)^2}}} =\frac{1}{3}*\frac{(-3x^2+2x+5)*\sqrt[3]{16x^2}}{\sqrt[3]{(-x^3+x^2+5x-6)^2}}$

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$4y^3+12xy^2 \frac{dy}{dx} =-3x^2+2x+5$


$\frac{dy}{dx} =\frac{-3x^2+2x+5-4y^3}{12xy^2}$

$y'=\frac{dy}{dx}$

$y' =\frac{-3x^2+2x+5-4y^3}{12xy^2}$