1. **State the problem:** We are given the implicit function $$x^2 - 3yx^2 + xyz - 2 = 0$$ and asked to find the partial derivative $$\frac{\partial z}{\partial y}$$. 2. **Recall the formula:** For implicit differentiation, treat $$z$$ as a function of $$y$$ (and $$x$$ as constant), then differentiate both sides with respect to $$y$$. 3. **Differentiate each term:** - $$\frac{\partial}{\partial y}(x^2) = 0$$ since $$x$$ is constant. - $$\frac{\partial}{\partial y}(-3yx^2) = -3x^2$$. - $$\frac{\partial}{\partial y}(xyz) = x \frac{\partial}{\partial y}(yz) = x\left(z + y \frac{\partial z}{\partial y}\right)$$ by product rule. - $$\frac{\partial}{\partial y}(-2) = 0$$. 4. **Write the differentiated equation:** $$0 - 3x^2 + x\left(z + y \frac{\partial z}{\partial y}\right) = 0$$ 5. **Simplify:** $$-3x^2 + xz + xy \frac{\partial z}{\partial y} = 0$$ 6. **Isolate $$\frac{\partial z}{\partial y}$$:** $$xy \frac{\partial z}{\partial y} = 3x^2 - xz$$ 7. **Divide both sides by $$xy$$:** $$\frac{\partial z}{\partial y} = \frac{3x^2 - xz}{xy} = \frac{3x - z}{y}$$ **Final answer:** $$\boxed{\frac{\partial z}{\partial y} = \frac{3x - z}{y}}$$