Concept: The magnitude of the derivative at a point is the slope of a tangent line at that point

Prerequisite concepts

Differential Calculus The derivative can be approximated by the slope of a secant line

The value of the derivative at a point on a curve can be approximated by the slope of a line secant with the curve near that point.

The magnitude of the derivative at a point is the slope of a tangent line at that point

Directional Derivatives

All partial derivatives can be found as a slope of a tangent plane

Just as the derivative of a one-dimensional function at a point is equal to the slope of a tangent line at that point, and partial derivative of a multivariable function can be found as a slope of a tangent plane.

The components of the gradient are partial derivatives

The Hot Plate

There is a partial derivative in every direction at any point There is a tangent line in every direction at every point The value of a partial derivative depends on the value(s) of what is held constant The magnitude of the gradient is proportional to the density of contour lines