Paper 4: Fundamentals of Business Mathematics & Statistic

3.32 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

Example 69: Given y= 2x 1x 2++ , find dydx.

Solution:

Let y u,= where

( ) ( ) ( ) ( )

( )^2

x 2 2x 1 2x 1 x 2d d

u 2x 1 du, dx dx

x 2 dx x 2

+ + + − + +

= + =

+

( ) ( )

( )^2 ( )^2

du x 2 .2 2x 1.1 3

dx x 2 x 2

= + − + =

+ +

dy 1 1 x 2.

du 2 u 2 2x 1

= = +

+

( )^2 ( )3/2

dy dy du 1 x 2 3. 3

dx du dx 2 2x 1 x 2 2 2x 1. x 2

= = + =

+ + + +

Example 70: If y = log log log x^2 , find dydx

Solution :

Let y = log u where u = log v and v = log x^2 = 2 logx.

2

dy 1 1 (^1).
du u logv= = =log logx
2
du 1 1 (^1).
dv v= =logx =2 logx
dv 2
dx x=
∴ 2 2
dy dy du dv.. (^1). 1 2. 1
dx du dv dx= =log logx 2 logx x=xlogxlog logx
Example 71: Differentiate x^5 w.r.t.x^2
Solution:
Let y = x^5 , z = x^2
dy 5x , 4 dz dx 2x, 2
dx= dx dx= = so that
dx 1.
dz 2x=
Now dy dy dxdz dx dz=. 5x .=^4 2x 21 5= x.^3