Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.39

SELF EXAMINATION QUESTIONS

Find

2

2

d y
dx^ in the following cases :
l. (i) y = 2x. (ii) y = 2x^4. (iii) y = 5x^3 -3x^2. (iv) y = (2 + 3x)^4.
[Ans. (i) 0 (ii) 24x^2 (iii) 30x - 6 (iv) 108 (2 + 3x)^2 ]

2. (i) y = x^4 e2x (ii) y = x (1 - x)^2 (iii) y = x^4 log x.
   [Ans. (i) 4x^2 e2x (x^2 + 4x + 3) (ii) 6x - 4 (iii) 7x^2 + 12x^3 .logx]
   (iii) 7x^2 + 12x^3 .locx)


3. (i) y=logxx (ii) y=1 x1 x+− [Ans. (i) 2log 3x 3 − (ii) 3

4

(1 x)− ]

4. (i ) x = at^2 , y = 2at, (ii)
   t^2 t
   x=1 t 1 t+ ,y= + (iii) x=1 t 2t1 t 1 t−+ ,y= +

[Ans. (i) 2at−^13 (ii) ( )

3

2 3

2(t 1)

t 2t

− +

+ (iii) 0]

5. If y = x^3 – 9x^2 + 9x find

2

2

d y

dx for x= 1, x = 3. [Ans. – 12, 0]

3.4.5 PARTIAL DERIVATIVE
We know that area of a rectangle is the product of its length and breadth, i.e., area = l × b. Now if the
length increases (when breadth is constant), area increases. If again breadth decreases (taking length as
constant), area decreases. So here area is a function of two independent variables, (i.e., length and breadth).

Again we know that area of a triangle is 21 base × altitude, i.e., area is a function of base and altitude,

(i.e., two variables).
If u is a function of two independent variables x and y, then we may write u = f(x, y).
The result obtained in differentiating u = f(x, y) w.r.t.x, treating y as a constant, is called the partial derivative

of u w.r.t.x and is denoted by any one of ∂∂u fx x. , f x, y ,u ,f∂∂ x( ) x x where

f

x

∂

∂ or

( ) ( )

x x 0

f limf x x,y f x, y,

∂ → x

= + ∂ −

∂ if it exists
Similarly, the partial derivative of u = f(x, y), w.r.t. y (treating x as a constant) is the result in differentiating u

= f(x, y), w.r.t. y and is denoted by ∂∂u fy y,∂∂ or fy where ∂∂yf or y y 0 ( )
f limf(x, y y) f x, y
∂ → y

= + ∂ −

∂ provided the limit
exists.
Note : The curl is used for partial derivative in order to make different form of symbol d of ordinary derivative.