The document outlines a presentation on differentiation given by six students - Muhammad Arsalan, Mian Ali Shafique, Ali Hassan, Awais Rao, Ahmed Rafique. It introduces the topic and provides an outline of key concepts to be covered including definition of a derivative, notation, constant rule, sum rule, definition rule, linear function rule, power function rule, product rule, quotient rule, chain rule, and inverse rule. Examples are given for many of the rules to demonstrate their use.

1. 17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 1

2. Differentiation
Presented by:
Muhammad Arsalan
Mian Ali Shafique( Group Leader)
Ali Hassan
Awais Rao
Ahmed Rafique
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 2

3. Differentiation
Course Outline:
Introduction
Definition
Notation
Constant Rule
Sum Rule
Definition Rule
Linear Function Rule
Power Function Rule
Product Rule
Quotient Rule
Chain Rule
Inverse Rule
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 3

4. Differentiation
Muhammad
Arsalan
17 March 2019
Arsalan Anjum From BS CS 1st In University
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5. Differentiation
Introduction:
It is our observation that, everything changes with
the passage of time.
In practical life, there occur many changes such as:
Speed of car
Inflation of currency
Number of bacteria in a culture
All these are related to derivative.
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 5

6. Differentiation
Definition:
In particular
The rate of change of a function with respect to the
independent variable is called derivative.
Other
Let f(x) be a function, then its denoted by f’(x)
Or d/dx f(x) and defined as;
f’(x)= lim hx>0 f(x+hx)-f(x)/hx
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 6

7. Differentiation
Notations:
Leibniz Newton Lagrance
dy/dx
or f (x) f’(x)
df/dx
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 7

8. Differentiation
Constant Rule:
The first rule of differentiation is that the
derivative of every constant function is
zero.
If y = c where c is a constant,
dy/dx=0
e.g. y = 10 then dy/dx=0
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 8

9. Differentiation
Mian Ali Shafique
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 9

10. Differentiation
Definition Rule:
For definition rule of f(x) four steps are used.
 Find f(x+hx) adding hx
 Simplify f(x+hx)-f(x)
 Divide f(x+hx)-f(x) by hx to get,
f(x+hx)-f(x)/hx and simplify it
 Apply limhx>0 f(x+hx)-f(x)/hx
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 10

11. Differentiation
Sum Rule:
If u and v are differentiable functions of x, then their
sum is differentiable
at every point where u and v are both differentiable.
At such points
d/dx(u+v)= d/dx(u) + d/dx(v)
e.g
d/dx(x4 + 10x)= d/dx(x4) + 10 d/dx(x)
= 4x3+ 10 (1)
d/dx(x4 + 10x) = 4x3 +10
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 11

12. Differentiation
Linear Function Rule:
If y = mx+b
dy/dx=m
e.g. y =6x+b
dy/dx=6 d/dx(x) + d/dx(b)
dy/dx=6(1) + 0
dy/dx=6
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 12

13. Differentiation
Ali Hassan
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 13

14. Differentiation
Power Function Rule:
If n is a negative integer and then
d/dx(xn)= nxn-1
e.g
Y = x3
dy/dx = d/dx(x3)
dy/dx = 3x3-1
dy/dx = 3x2
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 14

15. Differentiation
Product Rule:
The derivative of a product of two
functions is the sum of two products, as
we now explain.
If u and y are differentiable at x, then so
is their product uy, and
d/dx (u.v) = u d/dx (v) + v d/dx (u)
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 15

16. Differentiation
Product Rule:
e.g
y = (x-5)(x-7)
dy/dx = (x-5) d/dx (x-7) + (x-7) d/dx(x-5)
= (x-5) (1) + (x-7)(1)
= (x-5) + (x-7)
= (x-5+x-7)
dy/dx = 2x-12
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 16

17. Differentiation
Awais Rao
17 March 2019
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18. Differentiation
Quotient Rule:
If u and y are differentiable at x and if
then the quotient is differentiable at x,
and
d/dx (u)/(v)= u d/dx (v) – v d/dx (u)/(u)2
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 18

19. Differentiation
For Example:
y= (x+2)/(x+4)
dy/dx=(x+2)d/dx(x+4)-(x+4)d/dx(x+2)/(x+4)2
dy/dx= (x+2)(1) – (x+4)(1)/(x+4)2
dy/dx= (x+2-x-4)/(x+4)2
dy/dx= -2/(x+4)2
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 19

20. Differentiation
Chain Rule:
If y is a function of v, and v is a function of
x, then y is a function of x and
dy/dx= dy/du . Du/dx
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 20

21. Differentiation
For Example:
y = (4x3 + 3x – 7 )4 , v= 4x3 + 3x – 7 then y= v4
dy/dv=4v3 then dy/dv= 4(4x3 + 3x – 7 )3
dv/dx=d/dx(= 4x3 + 3x – 7 )then dv/dx=(12x2+3)
Using by chain rule
Dy/dx= dy/dv . dv/dx
Dy/dx= 4(4x3 + 3x – 7 )3 .(12x2+3)
17 March 2019
Arsalan Anjum From BS CS 1st In University
College of Management & Sciences Khanewal 21

22. Differentiation
Ahmed Rafique
17 March 2019
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23. Differentiation
Inverse Rule:
If x= f(y) then 1/dx/dy
Example:
(i) X=3y2
dx/dy= 6y then dy/dx= 1/6y
(ii) y=4x3
dy/dx= 12x2 then dx/dy= 1/12x2
17 March 2019
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24. End of Slides
17 March 2019
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