Chain Rule — Question 9

Question 9 – Solution

Strategy. Differentiate both constraints and solve the resulting $2\times 2$ linear systems.

Step 1: With respect to $s$ $$x_s+y_s=1,\qquad yx_s+xy_s=0.$$ Solving gives $$\boxed{x_s=\frac{x}{x-y}},\qquad\boxed{y_s=-\frac{y}{x-y}}.$$

Step 2: With respect to $t$ $$x_t+y_t=0,\qquad yx_t+xy_t=1,$$ so $$\boxed{x_t=\frac 1{y-x}},\qquad\boxed{y_t=\frac 1{x-y}}.$$ At $(1,2)$: $$\boxed{x_s=-1,\ y_s=2,\ x_t=1,\ y_t=-1}.$$

Step 3: Edge case The determinant of the coefficient matrix is $x-y$. When $x=y$, the system cannot locally distinguish the two roots; division by $x-y$ correctly signals failure of this local inverse description.