Example f(x, y) and g(u, v) using the matrix multiplication notation
Given
Missing \begin{bmatrix} or extra \end{bmatrix}2u - v \\ v - u \end{bmatrix}$$ and the outputs of $f \circ g$ at $u = 1, v = 2$ change at rate of $$\begin{bmatrix} -4 \\ 3 \end{bmatrix}$$ find rates of change of inputs $u, v$ **Solution** if $u = 1, v = 2$, then $x = 0, y = 1$ $$[Df]\biggr\rvert_{x=0, y=1} = \begin{bmatrix} 2x & 2y \\ y & x \end{bmatrix}_{x = 0, y= 1} = \begin{bmatrix} 0 & 2 \\ 1 & 0 \end{bmatrix}$$ and $$[Dg] = \begin{bmatrix} 2 & -1 \\ -1 & 1 \end{bmatrix}$$ (for any $u, v$) From chain rule,