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 f x and f y 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 =.

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 = and

Proof:

Special Cases: Where, u = <a, b> * If u = i = <1, 0>, then D i f = f x. ** If u = j = <0, 1>, then D j f = f y. 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 = and the formula becomes:

Example-1

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

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

Example-3:

Class work

proof

If we define a function g of the single variable h by then, by the definition of a derivative, we have the following equation.

On the other hand, we can write: g(h) = f(x, y) where: – x = x 0 + ha – y = y 0 + hb

Hence, the Chain Rule gives:

If we now put h = 0, then x = x 0 y = y 0 and

Comparing Equations 4 and 5, we see that:

Example-1

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