Paper 4: Fundamentals of Business Mathematics & Statistic

3.36 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

Example 76 : If y = x^4 , find

2

2

d y.

dx

Solution:

4 3 2 3 3 1 2

2

y x , 4x , againdy d y d dy d4x 4.3x 12x.

dx dx dx dx dx

= = =^ = = − =

Note : To find ( )

(^33222)
3 3 2
d y d y d d y d; 12x 12. xd
dx dx dx dx dx dx

=^ = =

(^)
= 12.2x2 – 1 = 24x i.e., third order derivative is 24x.
Example 77 : Find
2
2
d y
dx if
y logx.
= x
Solution:
1 2 2
x. logx.1^1
dy y x 1 logx;
dx x x

− −

= = =

2 2 ( )

2 2 3

x^1 1 logx. 2x

d y d dy d 1 logx x

dx dx dx dx x x

(^) − (^) − −
−
= = =

( ) ( )

3 2 2

x 2x 1 logx 1 2 1 logx 2 logx 3

x x x

=− − − =− − − = −.

Example 78 : If y = 5X, find

2

2

d y

dx

y =5x or, log y = x log 5

or,

(^1) , log5dy
y dx= or
dy y.log5
dx= or
dy 5x.log5
dx= .... (i)
Again dx dx dxd dy d^ = (5x.log5) or
(^2) x x
2
d y 5. log 5 log5 5d d
dx = dx + ⋅dx⋅
or,
(^2) x x
2
d y 5 .0 log5.log5.5
dx = +^ [from (i)]
= (log5)^2 .5x
Example 79 :
If y = x log x find
2
2
d y
dx