Directional derivatives

A directional derivative measures the local rate of change along a specified direction.

Different ways to understand it

Directional derivative
Khan Academy · English · Explanation
Connection evidence
Reviewed English material v1: card "Directional derivative as an extension of partial derivative" at 0:00 — The video introduces the directional derivative by first recalling partial derivatives for a scalar function of two variables. A partial derivative is tied to a coordinate direction: x for ∂f/∂x and y for the vertical analogue. The directional derivative keeps the same infinitesimal-comparison idea but allows the input nudge to point in any chosen direction in the plane.

Gradient
In Euclidean coordinates, partial derivatives form a vector describing directional change.
Multivariable functions
A multivariable function assigns an output to a tuple of inputs in its domain.
Antiderivatives
An antiderivative has the given function as its derivative; on an interval, any two antiderivatives differ by a constant.
Chain rule
The derivative of a differentiable composition combines the derivative of the outer function with that of the inner function.
Change of variables
A suitable coordinate change transforms volume elements through the absolute Jacobian determinant.