If $z = f(x − y)$ and $g(x, y) = x − y$, so that $z =f∘g$
Why/How does the chain rule imply this?
$$\frac{∂z}{∂x} =\frac{∂f}{∂g}\frac{∂g}{∂x} =\frac{∂f}{∂g}$$
and
$$\frac{∂z}{∂y}=\frac{∂f}{∂g}\frac{∂g}{∂y} =-\frac{∂f}{∂g}$$
2026-04-04 10:11:10.1775297470
Multivariable Chain Rule / Partial Derivatives
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Just note that $z = f(x-y)$ is one variable function depend on $g = x-y$. Because $z = f (g(x,y))$, $$ \frac{\partial z}{\partial x} = \frac{\partial f(g(x,y))}{\partial x} = \frac{\partial f}{\partial g} \frac{\partial g}{\partial x} = \frac{\partial f}{\partial g} \cdot 1 = \frac{\partial f}{\partial g} $$ and the same thing for $\frac{\partial z}{\partial y}$.