Chapter 15 - Differentiation in Several Variables - 15.6 The Chain Rule - Exercises - Page 809: 27

$$-\frac{2xy+z^2}{y^2+2xz}$$

Given $$x^{2} y+y^{2} z+x z^{2}=10$$ Consider $$F(x,y,z)=x^{2} y+y^{2} z+x z^{2}=10 $$ Then \begin{align*} F_x&= 2xy+z^2\\ F_z&= y^2+2xz \end{align*} Then \begin{align*} \frac{\partial z}{\partial x}&=-\frac{F_{x}}{F_{z}}\\ &=-\frac{2xy+z^2}{y^2+2xz} \end{align*}