# How to decompose change in xy?

Say I have the following equation:

$$z = x + y$$

Then we can say that:

$$\Delta z = \Delta x + \Delta y = (x_1 - x_0) + (y_1 - y_0)$$

Now, say that we have:

$$z = xy$$

How do I decompose this? It is clearly not the case that:

$$\Delta z = y_0\Delta x + x_0\Delta y$$

So how do I apportion where the changes are coming from?

For instance, say that $$x_0 = 2, x_1 = 4$$ and $$y_0 = 6, y_1 = 8$$.

Then, $$z_0 = 12, z_1 = 32$$.

How do I apportion how much of the change in $$z$$ of 20 is coming from $$x$$ vs $$y$$?

• $\Delta z=z-z_0=xy-x_0y_0=xy-xy_0+xy_0-x_0y_0=x\Delta y+y_0\Delta x=x_0\Delta y+y_0\Delta x+...$
– A.Γ.
Mar 1, 2020 at 18:08
• For reference, "Increment Theorem for Functions of Two Variables" Mar 1, 2020 at 18:13

Assuming $$f$$ is differentiable at $$(x,y)=(x_0, y_0)$$, $$\Delta z = f_x(x_0,y_0)\Delta x + f_y(x_0,y_0)\Delta + \epsilon_1\Delta x + \epsilon_2\Delta y$$ where
$$\epsilon_1 = f_x(c_x,y_0) -f_x(x_0,y_0)$$,
$$\epsilon_2 = f_y(x+\Delta x,d) -f_y(x_0,y_0)$$, you get $$c_x$$ and $$d$$ by MVT.
For small changes, you can ignore last two term to approximate $$\Delta z$$.