Page 250 - DMTH201_Basic Mathematics-1

Unit 9: Differential Calculus

f ( ) f ( ) Notes
x
a
f ( ) lim
a
x a x a
x
a
f ( ) f ( )
Notes f (a) exists if the lim exists as x a through values < a (left hand limit)
x a x a
f ( ) x f ( ) a
and lim through values > a (right hand limit) exist and further they are equal.
x a x a
9.1.1 Derivative of a Function - Method of First Principles

x
If y f ( ) is a function then as x changes y also changes.
A change in x is called the increment in x and is denoted by x. Corresponding change in y is
called increment in y and is denoted by y.
as x changes to x + x, y changes to y + y.

First Principles

Let y = f(x) ...(i)
y y ( f x ) x ...(ii)

Subtracting (i) from (ii), we get
x
y ( f x ) x f ( )
Divide both sides by x

x
y ( f x ) x f ( )
x x

Taking limits as x 0 , we get

y ( f x ) x f ( ) x
lim lim
x 0 x x 0 x

dy
If this limit exists then it is called the derivative of y w.r.t., x and is denoted by .
dx

y dy ( f x ) x f ( ) x
lim lim
x 0 x dx x 0 x
dy
is also called the differential coefficient of y w.r.t., x.
dx

dy
Notes should not be read as the product of d and y divided by the product of d and x.
dx
d
In fact, is the symbol for the derivative w.r.t. x or differential coefficient w.r.t. x.
dx

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