How does the directional derivative generalize the partial derivative from coordinate axes to an arbitrary vector direction?

Conditions

• The function is a scalar-valued function of two variables.
• The partial derivative is defined along coordinate axes.
• The directional derivative uses an arbitrary vector direction.

Reasoning, step by step

1. Start with the partial derivative as a horizontal or vertical nudge.
2. Introduce an arbitrary direction vector in the input plane.
3. Scale the direction vector by a small scalar h.
4. Take the limit as h approaches 0.
5. Compare the resulting output change to the infinitesimal input displacement.

Example

Instead of moving only right (x-direction) or only up (y-direction), the directional derivative considers moving left 1 and up 2, represented by the vector [-1, 2].

Common misconceptions

• Believing that the directional derivative is a completely new concept unrelated to partial derivatives.
• Thinking that the direction vector must be a unit vector for the generalization to hold.

Watch the explanation

Explore next

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