Property partial derivatives

Introduction: Similar as for functions of one variable, for functions of two variables we can approximate the change in the function value by the use of derivatives. Since we deal with two input variables, the value of both variables can be changed and we have to use partial derivatives.

Definition: Consider the function $z(x,y)$ at $(x_0,y_0)$. If $x_0$ changes with $\Delta x$ and $y_0$ changes with $\Delta y$, then the change in the function value can be approximated as follows:
$$\Delta z \approx z'_x(x_0,y_0) \Delta x + z'_y(x_0,y_0) \Delta y.$$

Remark 1: If the change $\Delta x$ or $\Delta y$ is positive, then $x$ or $y$ increases; when the change $\Delta x$ or $\Delta y$ is negative, then $x$ or $y$ decreases.

Remark 2: The smaller the changes in $x$ and/or $y$ the better the approximation.