Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.23

3.4 DERIVATIVE

Introduction :
The idea of limit as discussed previously will now be extended at present in determining the derivative of a
function f(x) with respect to x (the independent variable). For this let us know at first what the term ‘increment’
means.
Increment : By increment of a variable we mean the difference of initial value from the final value.
i.e., Increment = final value – initial value.
Let x change its value from 1 to 4, increment of x = 4 – 1 = 3.
Again if x changes from 1 to – 2, increment = – 2 – 1 = – 3.
(i.e., increment may be positive or negative).
Symbols : Increment of x will be denoted by h or, δx (delta x) or ∆x (delta x) and that of y will be represented
by k or, δy or, ∆y.
If in y = f(x), the independent variable x changes to x + δx, then increment of x = x + δx – x = δx (≠ 0).
So y = f(x) changes to y = f(x + δx).
∴ increment of y = f(x + δx) – f(x) [as y = f(x)]

Now the increment ratio δδyx=f x x f x f x h f x( + δδx)−( )= ( +h)−( ) [assuming δx =h]

If the ratio δδyxtends to a limit, as δx → 0 from either side, then this limit is known as the derivative of

y [ = f(x)] with respect to x.
Example : If y = 2x^2 , then y + δy = 2 (x + δx)^2 , δy = 2 (x + δx)^2 – 2x^2

y 2 x x 2x( )^22

x x

∴δ = + δ −

δ δ

Again for =x^15 , (^) ( )^55
y^11
δ = x x+ δ −x
Definition : A function y = f(x) is said to be derivable at x if δ →limx 0 δδyx
or, x 0 ( ) ( )
limf x x f x
δ → x

• δ −
  δ or,

( ) ( )

h 0

limf x h f x

→ h

+ −

exists and equal to l. Now this limit l that exists is known as derivative (or differential co-efficient) of y [or f(x)]
with respect to x.
Symbols : Derivative of y [ = f(x)] w.r.t.x (with respect to x) is denoted by

dy

dx or, f ′ (x), or, ( )

d f x
dx^
or, Dy or, D [f(x)] or, y 1