Natural Science Department – Duy Tan University

1. Natural Science Department – Duy Tan University Partial Derivatives In this section, we will learn: Various aspects of partial derivatives Lecturer: Ho Xuan Binh Da Nang-01/2015

2. 1 PARTIAL DERIVATIVES Natural Science Department – Duy Tan University If f is a function of two variables x and y, suppose we let only x vary while keeping y fixed, say y = b, where b is a constant. • Then, we are really considering a function of a single variable x:g(x) = f(x, b) Partial Derivatives

3. 1 PARTIAL DERIVATIVES Natural Science Department – Duy Tan University If g has a derivative at a, we call it the partialderivative of f with respect to xat (a, b). We denote it by: fx(a, b) Partial Derivatives

4. 1 PARTIAL DERIVATIVES Natural Science Department – Duy Tan University Similarly, the partial derivative of f with respect to yat(a, b), denoted by fy(a, b), is obtained by: Partial Derivatives

5. 1 PARTIAL DERIVATIVES Natural Science Department – Duy Tan University If f is a function of two variables, its partial derivativesare the functions fx and fy defined by: Partial Derivatives

6. 2 NOTATIONS FOR PARTIAL DERIVATIVES Natural Science Department – Duy Tan University If z = f(x, y), we write: Partial Derivatives

7. 3 RULE TO FIND PARTIAL DERIVATIVES OF z = f(x, y) Natural Science Department – Duy Tan University • To find fx, regard y as a constant and differentiate f(x, y) with respect to x. • To find fy, regard x as a constant and differentiate f(x, y) with respect to y. Partial Derivatives

8. 4 Example 1 Natural Science Department – Duy Tan University • If f(x, y) = x3 + x2y3 – 2y2find fx(2, 1) and fy(2, 1) Partial Derivatives

9. 5 Example 2 Natural Science Department – Duy Tan University • If • calculate Partial Derivatives

10. Thank you for your attention