Derivative By Muhammad Shahid DA SKBZ COLLEGE
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Derivative By
Muhammad ShahidDA SKBZ COLLEGE
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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
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Derivative What Does Derivative Mean?
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Derivative at a Point
axafxf
itafax
)()(im)(' l
axafxf
axaf
)()(lim)('
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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
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Application of derivative
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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
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Chain rule :-
dxdu
dudy
dxdy
dxdu
dudy
dxdy
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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
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Maximum
For a relative
maximum the value of
the function is increasing to the left
decreasing at the right
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Minimum
For a relative minimum the value of the function is decreasing to the left and increasing
the right
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Stationary point or critical point
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A point at which the concavity changes upward or down ward is called an inflection.
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Point of Inflection
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Practical implementation of derivative
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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
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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.
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