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 1 Write down all partial derivatives of the following function Solution First order PD

6. Second order PD

7. Second order PD (mixed partial)

8. In example 1, we observed that This properties hold for all functions provided that certain smoothness properties are satisfies. The mixed partial derivative must be equal whenever f is continuous.

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

10. Solution 2

11. Solution 2

12. 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

13. 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?

14. Solution 3 Volume of rectangular box: Possible error of the volume: y z

15. Solution 3 Possible error of the volume:

16. Solution 3 Surface Area of rectangular box: Possible error of the volume: y z

17. Solution 3 Possible error of the surface area:

18. 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.

19. Solution 4 Volume of cylinder: Rate of change:
(1)
(2)

20. Solution 4 From (1) Differentiate partially wrt t: Substitute in (2):

21. 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,

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

23. Solution 5
Let
Then,
Therefore ,

24. 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,

25. Example 6 Let , where and Find and

26. Solution 6

27. Solution 6 Therefore

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

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

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

31. Solution 7 Given and , determine the Jacobian for the system of equation.

32. 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:

33. Example 8 Given and , Find and

34. Solution 8 Given and , Find and

35. Example 9 Let and Find and

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

37. 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:

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

39. 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

40. Example 10 Find and classify the stationary points of

41. Solution 10 Find stationary point(s):

42. Substitute (2) in (1) Stationary points:

43. Find the Hessian function:

44. Determine the properties of SP:
Point
Hessian:
Conclusion
SP is a maximum point
SP is a saddle point