Finding the % Contribution Of Change In A Hyperbolic Parabola I've got the following equation, where $z$ and $x$ are both variables:
$$y = (zx)+(x+z)$$
What I want to know is: when z and x both change, what percentage of the change in y is driven by the change in z and what percentage is driven by the change in x?
You can stop reading here and answer that if you want, but I've provided my current though process and some examples below in case that helps:
If z and x were to change by equal amounts, then each variable should account for 50% of the total change in y since both z and x are performing the same operations. As an example:
Let's say that z and x both increase by 0.10. In that case, y would increase by 0.21.
Since  x and z each increased by the same amount and perform the same operations, then each variable contributed to 50% of the total change, or 0.105 points.
But is that true across all changes? If z increased by 0.2 and x increased by 0.1, did z account for 66% of the total change in y?
 A: Here is one "intuitive" idea: use directional derivatives.
Say you want to increase $x$ by $\Delta x = a$ and increase $z$ by $\Delta z = b$. Let $\vec{u} = \frac{1}{\sqrt{a^2+b^2}} (a,b)$ be the unit vector in the direction $(a,b)$. To make things simple, assume $(a,b)$ is a unit vector, so that $\sqrt{a^2+b^2} = 1$, and we won't have to have the square roots in our equations.
Then the directional derivative $D_{\vec{u}}y$ measures the infinitesimal change in $y$ as the point $(x,z)$ moves in the direction $(a,b)$. The formula for directional derivative is:
$$ D_{\vec{u}} y = a \frac{\partial y}{\partial x} + b \frac{\partial y}{\partial z} $$
You could say the "contribution" from changing $x$ is the term $a \frac{\partial y}{\partial x}$ and the contribution from changing $z$ is $b \frac{\partial y}{\partial z}$. Then one possible heuristic way of saying the "percent of total change coming from $x$" would be
$$ \frac{a \frac{\partial y}{\partial x}}{a \frac{\partial y}{\partial x} + b \frac{\partial y}{\partial z}} $$
and similarly for $z$.
In your example, you said increase $x$ by $0.1$ and $z$ by $0.2$. The unit vector $\vec{u}$ in this direction is $\vec{u} = \frac{1}{\sqrt{5}}(1,2)$. This means put $a = \frac{1}{\sqrt{5}}$ and $b=\frac{2}{\sqrt{5}}$ in the formulas above. Since $\frac{\partial y}{\partial x} = 1+z$ and $\frac{\partial y}{\partial z} = 1+x$, the directional derivative is:
$$ D_{\vec{u}} y = \frac{3+2x+z}{\sqrt{5}} $$
The percentage coming from $\Delta x$ is then $\frac{1+z}{3+2x+z}$, and the percentage coming from $\Delta z$ is $\frac{2(1+x)}{3+2x+z}$.
So you see, it's not enough to just say what are $\Delta x$ and $\Delta z$. The answer also depends on what point $(x,z)$ you start at.
