Question

A expressão da derivada y' para a função y = f(x) dada implicitamente pela expressão 6y² - cos y = 5 xy e?

Answer 1

$6y^2 - cos (y) = 5xy \\\\ \frac{d}{dx} (6y^2-cos(y)) = \frac{d}{dx} (5xy)\\\\ \frac{d}{dx} (6y^2) - \frac{d}{dx}(cos(y)) = \frac{d}{dx} (5xy)\\\\ 6\frac{d}{dx}(y^2) - \frac{d}{dx}(cos(y)) = 5\frac{d}{dx} (xy)\\\\ 6(2y\frac{dy}{dx} ) - (-sen(y)\frac{dy}{dx} ) = 5(\frac{dx}{dx} y + x\frac{dy}{dx} )\\\\ 12y\frac{dy}{dx} +sen(y)\frac{dy}{dx} = 5y+5x\frac{dy}{dx} \\\\ 12y\frac{dy}{dx} +sen(y)\frac{dy}{dx} - 5x\frac{dy}{dx} = 5y$

$\frac{dy}{dx}(12+sen(y)-5x) = 5y\\\\ \boxed{\boxed{\frac{dy}{dx} = \frac{5y}{12+sen(y)-5x} }}$