1. Da Nang-01/2015 Natural Science Department – Duy Tan University Lecturer: Ho Xuan Binh Directional Derivatives In this section, we will learn how to find: The rate of changes of a function of two or more variables in any direction.

2. Equations 1 0 Natural Science Department – Duy Tan University Directional Derivatives  Recall that, if z = f(x, y), then the partial derivatives f x and f y are defined as:

3. DIRECTIONAL DERIVATIVES 1 Natural Science Department – Duy Tan University Directional Derivatives 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.

4. DIRECTIONAL DERIVATIVES 1 Natural Science Department – Duy Tan University Directional Derivatives 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 =.

5. Definition 1 1 Natural Science Department – Duy Tan University Directional Derivatives  The directional derivative of f at (x 0, y 0 ) in the direction of a unit vector u = is: if this limit exists.

6. NOTATIONS FOR PARTIAL DERIVATIVES 2 Natural Science Department – Duy Tan University Directional Derivatives Comparing Definition 1 with Equations 1, we see that: If u = i =, then D i f = f x. If u = j =, 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.

7. Theorem 3 Natural Science Department – Duy Tan University Directional Derivatives 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

8. Example 4 Natural Science Department – Duy Tan University Directional Derivatives  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)?

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