1. CHAPTER 5 PARTIAL DERIVATIVES
INTRODUCTION
SMALL INCREMENTS & RATES OF CHANGE
IMPLICIT FUNCTIONS
CHAIN RULE
JACOBIAN FUNCTION
HESSIAN FUNCTION
STATIONARY POINT

2. INTRODUCTION Consider the following functions where
are independent variables.
If we differentiate f with respect variable , then we assume that
as a single variable
as constants

3. Notation If
First order partial derivatives:
Second order partial derivatives:

4. Example 1 Write down all partial derivatives of the following function

5. Example 2 Write down all partial derivatives of the following functions:

6. SMALL INCREMENTS & RATES OF CHANGE
Notation for small increment is
Let then
A small increment in z, is given by
Where are small increments of the stated variables
ii. Rate of change z wrt time, t is given by

7. Example 3 The measurements of closed rectangular box are length, x = 5m, width y = 3m, and height, z = 3.5m, with a possible error of in each measurement. What is the maximum possible error in the calculated value of the volume, V and the surface, S area of the box?

8. Example 4 The radius r of a cylinder is increasing at the rate of 0
Example 4 The radius r of a cylinder is increasing at the rate of 0.2cms-1 while the height, h is increasing at 0.5cms-1. Determine the rate of change for its volume when r=8cm and h =12cm.

9. IMPLICIT FUNCTIONS
Definition Let f be a function of two independent variables x and y, given by constant. To determine the derivative of this implicit function: Let Hence,

10. Example 5 Assume that y is a differentiable of x that satisfies the given function. Find using implicit differentiation.

11. THE CHAIN RULE
Definition Let z be a function of two independent variables x and y, while x and y are functions of two independent variables u and v. The derivatives of z with respect to u and v as follows: Hence,

12. Example 6 Let , where and Find and

13. JACOBIAN FUNCTION
Definition Let be n number of functions of n variables

14. JACOBIAN FUNCTION
Jacobian for this system of equations is given by: OR

15. Example 7 Given and , determine the Jacobian for the system of equation.

16. INVERSE FUNCTIONS FOR PARTIAL DERIVATIVES
Definition Let u and v be two functions of two independent variables x and y. . . Partial derivatives and are given by:

17. Example 8 Given and , Find and

18. Example 9 Let and Find and

19. HESSIAN FUNCTION
Definition Let f be a function of n number of variables . Hessian of f is given by the following determinant:

20. HESSIAN FUNCTION
Hessian of a function of 2 variables: Let f be a function of 2 independent variables x and y. Then the Hessian of f is given by:

21. HESSIAN FUNCTION
Stationary Point Definition Given a function . The stationary point of occurs when and Properties of Stationary Point

22. HESSIAN FUNCTION Properties of Stationary Point
If H<0, then stationary point is a SADDLE POINT
If H>0
MAXIMUM POINT if
MINIMUM POINT if
If H=0, then TEST FAILS or NO CONCLUSION

23. Example 10 Find and classify the stationary points of