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This lecture explains how to find the rate of change of a multivariable function in the direction of an arbitrary vector using the concept of directional derivatives. The lecture also discusses the maximum rate of change and the direction in which it occurs.
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⇤ I understand how to find the rate of change in any direction.
⇤ I understand in what direction the maximum rate of change happens.
Objectives
So far, we’ve learned the definition of the gradient vector and we know that it tells us the direc-
tion of steepest ascent. Its magnitude indicates the rate of change of the dependent variable in the
direction of the gradient.
A natural question to ask is, “What is the rate of change of the dependent variable in the direction
of an arbitrary vector ~v?” In other words, how fast does the surface ascend in the direction of ~v?
The rate of change in the direction of ~v is called the directional derivative. That is the scalar
projection of the gradient onto ~v.
Definition 16.1: The directional derivative, denoted Dv f (x, y), is a derivative of a multivariable
function in the direction of a vector ~v. It is the scalar projection of the gradient onto ~v.
Dv f (x, y) = compv rf (x, y) =
rf (x, y) · ~v
|~v|
It’s best to understand concepts with a picture. So let’s draw one. Consider the function
f (x, y) = x
2 y
2 .
The gradient of f is
rf (x, y) =
2 x