Question If -4y-y^2+x^3=-1-4y^3 then find (dy)/(dx) in terms of x and y. Answer Attemptiout of 3 (dy)/(dx)=

To find $\frac{dy}{dx}$, we need to differentiate both sides of the equation with respect to x. <br /><br />Differentiating the left side of the equation with respect to x, we get:<br /><br />$\frac{d}{dx}(-4y - y^2 + x^3) = -4\frac{dy}{dx} - 2y\frac{dy}{dx} + 3x^2$<br /><br />Differentiating the right side of the equation with respect to x, we get:<br /><br />$\frac{d}{dx}(-1 - 4y^3) = 0$<br /><br />Setting these two derivatives equal to each other, we have:<br /><br />$-4\frac{dy}{dx} - 2y\frac{dy}{dx} + 3x^2 = 0$<br /><br />Simplifying this equation, we get:<br /><br />$\frac{dy}{dx}(-4 - 2y) = -3x^2$<br /><br />Dividing both sides by $(-4 - 2y)$, we get:<br /><br />$\frac{dy}{dx} = \frac{-3x^2}{-4 - 2y}$<br /><br />Simplifying further, we get:<br /><br />$\frac{dy}{dx} = \frac{3x^2}{4 + 2y}$<br /><br /> the derivative $\frac{dy}{dx}$ in terms of x and y is $\frac{3x^2}{4 + 2y}$.