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Choose the direction and a point on the two-dimensional contour map in which you want to determine the partial derivative. For example, if you want to estimate the partial derivative in the x-axis, you will consider only the changes to the map in the horizontal direction surrounding the chosen point.
Choose two points which lie on the same horizontal plane as the point chosen, that are equidistant from the location at which you are estimating the partial derivative. For example, if the location at which you are estimating the derivative is (20, 20), the two points you choose to estimate the derivative could be located at (15, 20) and (25, 20), for example.
Find the elevation of the contours located at the two points chosen. These elevations are generally already defined on
the contour map. As an example, assume the elevation at (15, 20) is 300 and the elevation at (25, 20) is 400.
Subtract the difference in locations of the two points in the desired direction. In the example above, this means subtracting the horizontal positions of the two positions, which gives 25 - 15, which is 10.
Subtract the elevation at the two positions. In this example, the heights that correspond to the positions (25, 20) and (15, 20) are 400 and 300, so subtracting these elevations gives 100.
Divide the difference in the elevations by the difference in horizontal position on the map. Concluding the example used here, this gives 100 divided by 10, which gives 10. Therefore, the height changes by 10 units for every horizontal unit traveled for the chosen location on the contour map.