1. 1 Supplement Partial Derivatives

2. 2 1-D Derivatives (Review) Given f(x). f'(x) = df/dx = Rate of change of f at point x w.r.t. x h f(x + h) – f(x) xx+h

3. 3 1 st -order Partial Derivatives For a function with 2 variables, f(x, y)

4. 4 Evaluating Partial Derivatives To find f x, we differentiate f w.r.t. x and treat all other variables as constants. Example 1: f(x, y) = x 3 y 2 = (y 2 )x 3 To evaluate f x, y 2 is to be treated as a constant term. Thus f x = (y 2 )(3x 2 ) = 3x 2 y 2 Example 2: f(x, y, z) = 10xyz + sin(y)e x z 3 = (10yz)x + (sin(y)z 3 )e x To evaluate f x, (10yz) and (sin(y)z 3 ) are to be treated as constant terms. Thus f x = 10yz + (sin(y)z 3 )e x

5. 5 Exercise 1) f(x, y) = xe xy f x = f y = 2) f(x, y, z) = sin (xy) f x = f y = f z =

6. 6 2 nd -order Partial Derivatives

7. 7 Exercise f(x, y) = x 3 + x 2 y – 3x y 2 + y 3 f x = f y = f xy = f yx =