Activity: Directional Derivatives

Prerequisite concepts

Differential Calculus The derivative is a linear operator

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The Heater II

There is a partial derivative in every direction at any point

For a smooth, two dimensional function, a partial derivative can be found at any point and in any direction.

Derivatives can be found while holding one or more variables constant

Chain Rule

A partial derivative can be expressed in terms of other partial derivatives

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Instructor guide mvdderiv

Directional Derivatives

This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.

Representations used

$\frac{\partial f}{\partial x}$

Inclinometer

$\vec \nabla f$

Concepts taught

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

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