To find $\frac{dy}{dx}$ using implicit differentiation for the equation:

$x^2(x + y) = y^2(4x - y)$

we will differentiate both sides of the equation with respect to $x$.

Step 1: Differentiate Both Sides

The equation is:

$x^2(x + y) = y^2(4x - y)$

Differentiating both sides with respect to $x$:

Left Side

For the left side, we have $x^2(x + y)$. We can apply the product rule here.

$\frac{d}{dx} \left( x^2(x + y) \right) = \frac{d}{dx}(x^2) \cdot (x + y) + x^2 \cdot \frac{d}{dx}(x + y)$

1. $\frac{d}{dx}(x^2) = 2x$, so ( 2x(x + y