Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.33

Example 72: y = loge (^) (x x a ,+^2 +^2 ) find dydx.
Solution:
Let y = log u where u = x + x a^2 +^2

( )

du d dx x a 2 2 1/2

dx dx dx= + + or^22

du 1 1 .2x

dx= +2 x a+

Now

2 2

2 2 2 2 2 2 2 2

dy dy du 1. 1 x (^1) .x x a 1
dx du dx u x a x x a x a x a

= =^ +^ = + + =

(^) + (^) + + + +.

SELF EXAMINATION QUESTIONS

Differentiate the following functions w.r.t.x :

1. (x^2 + 5)^2. [Ans. 4x (x^2 + 5)]


2. (i) (ax + b)^5. [Ans. 5a (ax + b)^4 ] (ii) (1 – 5x)^6. [Ans. – 30 (1 – 5x)^5 ]

(iii) (3 – 5x)3/2. [Ans. −^125 3 5x− ]

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


4. 3x 7^2 + [Ans. 2

3 x

3x 7− ]

5. (2x^2 + 5x – 7)– 2 [Ans. – 2 (4x + 5) (2x^2 + 5x –7)– 3]


6. x 1 x.^3 −^2 [Ans.
   2 2 4
   2
   3x 1 x x
   1 x

− −

− ]

7.

2

2

x 1

x 1

−

+ [Ans. ( )( )

2 2 3/2

2x

x 1 x 1− + ]

8. e4x [Ans. 4e4x]


9. (i) e3x 4x 7^2 + − [Ans. (6x + 4)e3x 4x 7^2 + − ]

(ii) e3x 6x 2^2 − + [Ans. 6 (x – 1) e3x 6x 2^2 − + ]

10. log (x^2 + 2x + 5). [Ans. 2 ( )
    2 x 1
    x 2x 5

+

+ + ]

11. log x 1 x 1 .( + − − ) [Ans. 2

1

2 x 1

−

− ]