Interpretations of Partial Derivatives

A surface defined by a function of two variables can be visualized as a vast, uneven terrain, where each point is identified using Cartesian coordinates. The elevation of the terrain at any point is determined by a function that assigns a height value to every pair of horizontal coordinates. This representation allows the surface to be studied in terms of how its height varies across different directions.

At a specific point on this terrain, understanding how the height changes requires examining movement along particular directions. If motion is restricted along the x-direction while keeping the y-coordinate fixed, the resulting change in elevation describes the steepness in that direction. Similarly, movement along the y-direction, with the x-coordinate held constant, reveals the steepness in the perpendicular direction. These directional slopes indicate whether the terrain rises or falls: positive values correspond to upward movement, while negative values indicate a downward slope.

Although these directional measures provide useful insights, they represent only individual slices of the surface. When considered together, they define a planar surface that closely approximates the terrain at the given point. This plane, known as the tangent plane, touches the surface at that point and reflects its overall tilt. It serves as a local linear approximation, capturing how the surface behaves in the immediate vicinity and providing a simplified representation of the terrain’s geometry.