Directional Derivatives ExampleWhats the slope of at 0

Directional Derivatives

Example…What’s the slope of at (0, 1/2)? What’s wrong with the way the question is posed? What’s the slope along the direction of the x-axis? What’s the slope along the direction of the y-axis?

Recall that, if z = f(x, y), then the partial derivatives fx and fy are defined as: They represent the rates of change of z in the x- and y-directions—that is, in the directions of the unit vectors i and j.

• Suppose that we now wish to find the rate of change of z at (x 0, y 0) in the direction of an arbitrary unit vector u = <a, b>.

The rate of change of f(x, y) in the direction of the unit vector is called the directional derivative and is denoted by

Theorem: If f is a differentiable function of x and y, then f has a directional derivative in the direction of any unit vector u = <a, b> and

Proof:

Special Cases: Where, u = <a, b> * If u = i = <1, 0>, then Di f = fx. ** If u = j = <0, 1>, then Dj f = fy. In other words, the partial derivatives of f with respect to x and y are just special cases of the directional derivative.

• Suppose the unit vector u makes an angle θ with the positive x-axis, as shown.

• Then, we can write u = <cos θ, sin θ> and the formula becomes:

Example-1

Example-2: Find the directional derivative Duf(x, y) if: – f(x, y) = x 3 – 3 xy + 4 y 2 – u is the unit vector given by angle What is Duf (1, 2)? θ = π/6

How about the directional derivative for a function of 3 variables?

Class work-1

Class work-2: Find where in the direction of

Quick survey I feel I understand “DD” a) Very well b) With some review, I’ll be good c) Not really d) Not at all