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

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

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

Example 1 Write down all partial derivatives of the following function

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

Second order PD

Second order PD (mixed partial)

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.

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

Solution 2

Solution 2

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

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?

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

Solution 3 Possible error of the volume:

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

Solution 3 Possible error of the surface area:

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.

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

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

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,

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

Solution 5 Let Then, Therefore ,

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,

Example 6 Let , where and Find and

Solution 6

Solution 6 Therefore

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

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

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

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

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:

Example 8 Given and , Find and

Solution 8 Given and , Find and

Example 9 Let and Find and

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

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:

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

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

Example 10 Find and classify the stationary points of

Solution 10 Find stationary point(s):

Substitute (2) in (1) Stationary points:

Find the Hessian function:

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