Derivative By

Muhammad ShahidDA SKBZ COLLEGE

CONTENTS1)What is Derivative2)Derivative at a point3)Example4)Instantaneous Rate of Change5)Application of Derivative6)Example7)Chain Rule8)Example of Chain Rule9)Higher order Derivative10)Example of Maximum and Minimum

Derivative What Does Derivative Mean?

Derivative at a Point

axafxf

itafax

)()(im)(' l

axafxf

axaf

)()(lim)('

Instantaneous rate of change

EXAMPLE: Suppose that we are interested in determining how fast car is moving at the instant t=1. We might determine the instantaneous velocity by examining the average velocity during the interval near t=1

Application of derivative

Example:2

A flu epidemic is spreading through a large Midwestern stage. Based upon similar epidemic which have occurred in the past epidemiologist have formulated a mathematical function which estimate the number of person who will be afflicted by the flu , the function is

n=f(t)= -0.3t³+10t²+300t+250a)Find the instantaneous rate at which flu is expected to be spreading at t=11 and t=12 day

Chain rule :-

dxdu

dudy

dxdy

dxdu

dudy

dxdy

Higher order derivative

The 1st derivatives a measure of the instantaneous rate of change in the value of y with respect to a change in x the 2nd derivatives is a measure of the instantaneous rate of change in the slop with respect to a change in x

Maximum

For a relative

maximum the value of

the function is increasing to the left

decreasing at the right

Minimum

For a relative minimum the value of the function is decreasing to the left and increasing

the right

Stationary point or critical point

A point at which the concavity changes upward or down ward is called an inflection. 

Point of Inflection

Practical implementation of derivative

Example: An electric current when flows in a circular coil of radius “r” exerts a force

on a small magnet located at a distance “x” above the centre of the coil show that “F” is maximum when x=r/2

25

22 )( rx

kx

25

22 )( rx

kxF

Example : 8 A person wishes to fence in a rectangular garden which is to have an area of 1500 sq feet. Determine the dimensions which will create the desired area will require the minimum length of fencing Sol :- let x be the length and y be the width of the given rectangle Area = xy 1500 = xy Perimeter of rectangle = 2 ( x+ y ) y P = 2 ( x + x P = 2x + 3000x-1

P’ = 2 - 3000x-2

P” = 6000 x-3

P’ = 0 2 -  x2 = 1500  x = 38. 73 + t P” (38 . 73) = = 0.1 > 0   Relative minimum Y = Y = 38 .73 ft.