1. **State the problem:** Given the implicit equation $$x^2 z^4 + y^3 z^5 = x + y,$$ find the partial derivatives $$z_x = \frac{\partial z}{\partial x}$$ and $$y z_y = y \frac{\partial z}{\partial y}$$. 2. **Recall the implicit differentiation formula:** For an equation involving $$x, y, z$$, where $$z$$ is implicitly a function of $$x$$ and $$y$$, differentiate both sides with respect to $$x$$ and $$y$$ treating $$z$$ as a function of those variables. 3. **Differentiate with respect to $$x$$:** $$\frac{\partial}{\partial x} \left(x^2 z^4 + y^3 z^5\right) = \frac{\partial}{\partial x} (x + y)$$ Using the product and chain rules: $$2x z^4 + x^2 \cdot 4 z^3 z_x + 0 + y^3 \cdot 5 z^4 z_x = 1 + 0$$ Simplify: $$2x z^4 + (4 x^2 z^3 + 5 y^3 z^4) z_x = 1$$ 4. **Solve for $$z_x$$:** $$ (4 x^2 z^3 + 5 y^3 z^4) z_x = 1 - 2x z^4 $$ $$ z_x = \frac{1 - 2x z^4}{4 x^2 z^3 + 5 y^3 z^4} $$ 5. **Differentiate with respect to $$y$$:** $$\frac{\partial}{\partial y} \left(x^2 z^4 + y^3 z^5\right) = \frac{\partial}{\partial y} (x + y)$$ Using product and chain rules: $$0 + 3 y^2 z^5 + y^3 \cdot 5 z^4 z_y = 0 + 1$$ Simplify: $$3 y^2 z^5 + 5 y^3 z^4 z_y = 1$$ 6. **Solve for $$y z_y$$:** $$5 y^3 z^4 z_y = 1 - 3 y^2 z^5$$ Divide both sides by $$5 y^3 z^4$$: $$z_y = \frac{1 - 3 y^2 z^5}{5 y^3 z^4}$$ Multiply both sides by $$y$$: $$y z_y = y \cdot \frac{1 - 3 y^2 z^5}{5 y^3 z^4} = \frac{y - 3 y^3 z^5}{5 y^3 z^4}$$ Cancel common factor $$y$$ in numerator and denominator: $$y z_y = \frac{\cancel{y} - 3 y^3 z^5}{5 \cancel{y} y^2 z^4} = \frac{1 - 3 y^2 z^5}{5 y^2 z^4}$$ **Final answers:** $$z_x = \frac{1 - 2x z^4}{4 x^2 z^3 + 5 y^3 z^4}$$ $$y z_y = \frac{1 - 3 y^2 z^5}{5 y^2 z^4}$$