Directional Derivatives

Given a function in 3-dimensional space and a point in which that function exists:

The directional derivative () of function () at point in the direction of a unit vector () is defined by:

But what does this mean? If you recall that the derivative of a function is merely the line tangent to the function, but when working with solids we actually find a tangent plane, not a line (as stated in partial derivatives). But we can still find a line tangent to the solid, it just has to be one this plane we found. To specify what direction along this line we want to use we use the unit vector as listed above. See Multivariable Calculus Notes - Chapter 14 - Sections 5-6.

If we know the gradient function of the function we are trying to find the directional derivative of we can use the following theorem to find the directional derivative:

Where "" is the dot product and is the unit vector. See gradient notation for notation information. See Multivariable Calculus Notes - Chapter 14 - Sections 5-6.

Maximizing the Directional Derivative

Imagine the tangent plane to a solid, now imagine you drew two, perpendicular lines along this plane. If you were to measure their slopes the would be different (when the plane is not horizontal or vertical of course).

Image Credit: Alexwright, for Wikipeida (Modified)

In the above image the red line will have a greater rate of change than that of the blue line. But how do you figure out which line along that plane has the highest rate of change?

Given a point along the original function then:

The function increases the most rapidly in the direction of the gradient vector and the maximum rate of change is the magnitude of the gradient vector, .
The function decreases the most rapidly in the direction opposite of the gradient vector and the minimum rate of change is .

See Multivariable Calculus Notes - Chapter 14 - Sections5-6 for more information.