Concept: Partial derivatives are local quantities

Prerequisite concepts

Multivariable Calculus The derivative is a local quantity

The value of the derivative does not depend on what is happening far away from the point at which the derivative is being evaluated: it can be determined by what is happening in the neighborhood of the point. Geometrically, the derivative (slope) of a curve at a point is not a statement about the general shape of the curve but rather about the behavior of the curve immediately around that point.

Partial derivatives are local quantities

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