Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.25

Example 50 : (^) dxd(x 4x ;^4 )=^3 dxdx 1.x= 1 1− = 1x^0 = 1.1 = 1.
d 1 d x 1.x (^111) 1x 2 21
dx x dx x
(^) = − = − − − = − − =−
d. x dx^121 .x1/2 (^1) ;
dx dx (^2) 2 x

= = − =

( )

(^12) 3/2
3/2
d 1 d. .x (^1) .x 1 d; x x
dx x dx (^2) 2x dx

= − = − − = −

d.x 3 /2 (^3) x^123 x ;
=dx = 2 = 2

SELF EXAMINATION QUESTIONS

Examples 51: dxd(x x ) x^3 +^2 =dx dxd^3 + dx 3x 2x.^2 =^2 =

Example 52 : dxd(x .e ) x e e x x e e .2x x e 2xe .2 x =^2 dxd x+ xdxd^2 = 2 x+ x = 2 x+ x

Example 53:
d 4 dx^433
dx(2x ) 2= dx =2.4x 8x .=

Example 54 : If
x^2
y=x 1+. Let y=uv where u = x^2 , dx dxdu d= (x 2x 2x^2 )= 2 1− =

And v x 1 ,=( +) dx dxdv d= (x 1+)=dx dxd dx .1 1 0 1+ = + =

Now ( )
( ) ( ) ( )

(^2222)
2 2 2 2
v u.du dv
dy d u dx dx x 1.2x x .1 2x 2x x x 2x
dx dx v v x 1 x 1 x 1

− + − + − +

= = = = =

+ + +

Example 55: Find dydx of the following functions :

(i) x^4 + 4x, (ii) 3x^5 – 5x^3 + 110, (iii) – 2 + (4/5) x^5 – (7/8) x^8.

Solution:
(i) Let y = x^4 + 4x.

dy ddx dx= (x 4x (^4) + )=dx dxdx (^4) + d(4x 4.x 4 .x 4x 4.)= 4 1− + dxd = (^3) +
(ii) Let y = 3x^5 – 5x^3 + 110