3.34 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS
Calculus
- log log log x^2. [Ans. 2 2
(^1). 1 2.
log logx logx x]
- If y 1 x ,= +^2 prove that y x.ddxy=
- Differentiate x^6 w.r.t.x^4. [Ans.^32 x^2 ]
- 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