Paper 4: Fundamentals of Business Mathematics & Statistic

3.34 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

12. log log log x^2. [Ans. 2 2

(^1). 1 2.
log logx logx x]

13. If y 1 x ,= +^2 prove that y x.ddxy=


14. Differentiate x^6 w.r.t.x^4. [Ans.^32 x^2 ]


15. Differentiate x^5 w.r.t.x^2 [Ans.^52 x^3 ]
    3.4.2 DERIVATIVE OF IMPLICIT FUNCTION
    If f(x, y) = 0 defines y as a derivable function of x, then differentiate each term w.r.t.x. The idea will be clear
    from the given example.

Example 73 : Find dydx, if 3x^4 – x^2 y + 2y^3 = 0

Solution:

Differentiating each term of the functions w.r.t.x we get, 3.4x^3 –^22

x dy 2xy 6y dy 0

dx dx

(^) + (^) + =
(^)
or, 12x x^3 −^2 dydx−2xy 6y+^2 dxdy=^0
or, (6y x^2 −^2 )dydx=2xy 12x−^3 or,
3
2 2
dy 2xy 12x
dx 6y x

= −

−.

3.4.3 DERIVATIVE OF PARAMETRIC FUNCTION

Each of variable x and y can be expressed in terms of a third variable (known as parametric function). For

example. X = f 1 (t), y = f 2 (t). Now to find dydx we are to find dydt and dxdt so that dy dy dt dy /dt dxdx dt dx dx /dt dt=. = , 0.≠

Example 74 : Find dydxwhen x = 4t, y = 2t^2

Solution:

dx dy dy dy /dt 4t4, 4t, t

dt dt dx dx /dt 4= = = = =

Example 75 : Find dydx, when x=1 t3 at+ 3 ,

2

3

y 3at

=1 t+

Solution:

( ) ( )

( )

( )

( )

3 2 2

32 32

dx 1 t .3a 3at 3t 3a 1 2t

dt 1 t 1 t

+ − −

= =

+ +