Page 13 - (1-1)eng

(1 - 1) The First Derivative of a Function
Rules of Differentiation

The first derivative of the function y =  ( x ) is denoted by any of the following symbols

dy d d

, ( y ) ,  ( x ) ,  '( x ) , y'
dx dx dx

1) Constant function y = c , c   y' = 0
y  1  dy   1 2
2) Power rule y = x , n   y' = n x n – 1 x dx x 1
n
dy
y  x  
dx 2 x
3) Constant × function y = c × ƒ( x), c   y' = c × ƒ' ( x )

4) Sum or difference y = ƒ ( x ) ± g ( x ) y' = ƒ' ± g'
5) Product rule y = ƒ ( x )  g ( x ) y' = ƒ' g + g' ƒ
6) Quotient rule ƒ x ƒ' g  g ƒ '
 
y  y' =
 
g x g 2

Example (1)
Find the first derivative for each of the following functions:

1) y = 3
4
2) y = x
3) y = 5 x 7
3
4) y = 2 x – 3 x + 5
2
5) y = ( 3 x + 1 ) ( 5 x + 2 )
2 x
6) y =
3x 5

Solution:
1) y' = 0
3
2) y' = 4 x
6
3) y' = 5  7 x = 35 x 6
2
4) y' = 6 x – 3
2
2
5) y' = ( 6 x )( 5 x + 2 ) + ( 5 ) ( 3 x + 1 ) = 45 x + 12 x + 5

2 3x 5  3 2 x  10
6) y' = 2 = 2
3x 5  3x 5 

Calculus 12 Unit (1)