Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.29

Part B Regarding product functions :
Differentiate the following with respect to x

11. (i) x (x + 1) (ii)x^2 (x - 1) [Ans. (i) 2x + 1 (ii) 3x^2 - 2x]


12. (i) (x + 1) (x + 2) (ii)(x^2 + 1) x^2 [Ans. (i) 2x + 3 (ii) 4x^3 + 2x]


13. (i) (x - 1 )^2 (x + 2) (ii)(x^2 + 1) (x - 2)^3. [Ans. (i) 3x^2 - 3 (ii) (x - 2)^2 ( 5x^2 - 4x + 3)]


14. (i) x. ex (ii)x^10 ex (iii)ex. logx (iv)2ex (logx + 4).

[Ans. (i) x.ex + ex (ii)xl0ex+ lOxV (iii) ex.^1 x + exlogx (iv) 2ex.^1 x + 2ex (logx + 4)]

16. (i) (x+ l)(x + 2)(x + 4) (ii)(x^2 + 1) (x+ 2^2 ) (x^2 -2) [Ans. (i) 3x^2 + 14x + 14 (ii) 5x^4 + 16x^3 - 3x^2 - 8x - 2]
    Part C Regarding division of functions :
    Differentiate the following with respect to x.


17. (i) 1 x+x 2 (ii)
    x^2
    1 x+ (iii)

x 1

x 2

+

+ (iv)^2

x 4

x 2

+

+

[Ans. (i) ( )

2

22

1 x

1 x

−

+ (ii) ( )

2

2

2x x

1 x

+

+ (iii) ( )^2

1

x 2+ (iv)

( )

( )

2

2 2

x 8x 2

x 2

− + −

+

18. (i)
    x 1^2
    x 1

+

− (ii)^3

x 1

x 1

+

− (iii)

( )

( )

2

3

2 5x

x 1

−

− (iv)

4

4

x 1

x 1

−

+

[Ans. (i)

2

2

x 2x 1

(x 1)

− −

− (ii)

(^32 )

3 2

2x 3x 1

(x 1)

− + +

−

(iii) ( )

4 3 2

3 2

25x 40x 12x 50x 20

x 1

− + − − +

− (iv) ( )

4

4 2

8x

x 1+ ]

19. (i)
    x^2
    logx (ii)^2

logx

x (iii)

ex

x (iv)

2

x

x

e

[Ans. (i) ( )^2

2x logx x

logx

−

(ii) x 2x logx− x 4 (iii) ( )

x

2

e x 1

x

−

(iv) ( )

x

2x

xe 2 x

e

−

]

3.4.1 DERIVATIVE OF FUNCTION OF A FUNCTION

A variable y may be a function of a second variable z, which again may be a function of a third variable
x.
i.e., y = z^2 + 3, and z = 2x + 1
Here y is a function of z and z again a function of x. Ultimately y is seen to depend on x, so y is called the
function of another function.
Symbolically, if y = f (z), z = φ (x) then y = f {φ (x)}