Page 26 - (1-1)eng

(1 - 1) The First Derivative of a Function – G(C) A1

Group (C)

Answer the following questions
 
23) If  ( x ) = g x where g and h are two differentiable functions at x = a and
 
h x

g'   a
' ( a ) = 0, prove that:  ( a ) = .
h'   a

3
2
3
24) If  ( x ) + ' ( x ) = x + 5 x + x + 2, find  ( x ).      x  x  2x  3x 5 
2
 
x 1
25) If y = ƒ [ g ( x ) ], ƒ' ( x ) = and g ( x ) = , prove that:
x  1 x
2
 1 x  0



dy  x 2 1 x 2
 
dx   1 x  0
 x 2 1 x 2


2
26) Write what is equal to ( 1 + x + x + x + …+ x ) and by using differentiation,
3
n

n x n 1   n 1 x  n  1
2
prove that: 1 + 2 x + 3 x +…+ n x n – 1 = .
 x 1  2

2 x 1

27) If  ( x ) = 2 x 1 , prove that ' ( x ) = .
2 x 1  2 x 1


S.B. 28) ABC is a triangle of area S and C is a point moving on the straight line y = 2 x. If
A ( L, 0 ), B ( 0, K ) where L and K are two positive constants, prove that

dS = L + K provided that x > LK
dx 2 2 L  K

 dy  y 2
29) If y = x   x 2 1  , prove that x 2 1   
x
 dx  x 2

30) Find the interval to which a belongs such that the first derivative of the function

2
3
 ( x ) = x 2 a x 4  is defined on  and find the value of the first
derivative at the middle of the interval of a.   a    4 , 4  , '   x 3 4 x  
4
  3 x 2   

Calculus 25 Unit (1)