Question

Encuemtre dy/dx por medio de la diferencia implicita. x^1/5 + y^1/5=4

Answer 1

Derivemos ambos miembros

$\dfrac{d\left (x^{1/5}+y^{1/5}\right)}{dx}=\dfrac{d}{dx}(4)\\ \\ \\ \dfrac{d(x^{1/5})}{dx}+\dfrac{d(y^{1/5})}{dx}=0\\ \\ \\ \dfrac{1}{5}x^{\frac{1}{5}-1}+\dfrac{1}{5}y^{\frac{1}{5}-1}\cdot\dfrac{dy}{dx}=0\\ \\ \\ \dfrac{1}{5}x^{-4/5}+\dfrac{1}{5}y^{-4/5}\cdot\dfrac{dy}{dx}=0\\ \\ \\ \dfrac{1}{5}y^{-4/5}\cdot\dfrac{dy}{dx}=-\dfrac{1}{5}x^{-4/5}\\ \\ \\ \dfrac{dy}{dx}=-\dfrac{x^{-4/5}}{y^{-4/5}}\\ \\ \\ \dfrac{dy}{dx}=-\dfrac{y^{4/5}}{x^{4/5}}$


$\boxed{\dfrac{dy}{dx}=-\left( \dfrac{y}{x}\right)^{4/5}}$