Question

Achar a derivada y' = dy / dx da funcao implicita y: tg y = xy

Answer 1

$\mathrm{tg\,}y=xy$

Sabendo que $y$ é uma função de $x,$ vamos derivar os dois lados da igualdade acima:

$\dfrac{d}{dx}(\mathrm{tg\,}y)=\dfrac{d}{dx}(xy)\\\\\\ \sec^2 y\cdot \dfrac{dy}{dx}=\dfrac{d}{dx}(x)\cdot y+x\cdot \dfrac{dy}{dx}\\\\\\ \sec^2 y\cdot \dfrac{dy}{dx}=1\cdot y+x\cdot \dfrac{dy}{dx}\\\\\\ \sec^2 y\cdot \dfrac{dy}{dx}-x\cdot \dfrac{dy}{dx}=y\\\\\\ (\sec^2 y-x)\cdot \dfrac{dy}{dx}=y\\\\\\ \therefore~~\boxed{\begin{array}{c} \dfrac{dy}{dx}=\dfrac{y}{\sec^2 y-x} \end{array}}$