What is the geometric meaning of the directional derivative of a scalar function of two variables at a point in an arbitrary direction?

Conditions

• The function is a scalar-valued function of two variables.
• The direction is specified by a vector in the input plane.
• The step size approaches zero.

Reasoning, step by step

1. Identify the base point in the input plane.
2. Define an arbitrary direction vector.
3. Scale the direction vector by a small scalar h.
4. Take the limit as h approaches 0.
5. Interpret the result as the output change relative to the input displacement.

Example

For a function $f(x,y)$ and direction vector $v=[-1,2]$, the directional derivative measures the output change when moving slightly left and up from the point (1,2).

Common misconceptions

• Confusing the direction vector with the actual infinitesimal displacement.
• Thinking the directional derivative is the slope of the surface in 3D space rather than the slope of the 2D section curve.

Watch the explanation

Explore next

• How does the directional derivative differ from a partial derivative?
• Why does the video use $h*v$ instead of just v when defining the directional derivative?
• What is the formula for the directional derivative in direction $w=(a,b)$?