Page 18 - (1-1)eng

(1 - 1) The First Derivative of a Function

n
8) Power of a function y = [ ƒ ( x ) ] dy  n   x  n 1  ' x y  1  dy   '   x
 

dx    x dx    x   2

 
dy  ' x
y      
x
dx 2   
x
Example (9)
dy
Find if
dx
10
3
(a) y = ( 6 x + 3 x + 1 ) (b) y = x 2 3x 1
 x 2  x 2  9
(c) y = (d) y =
x  x    1 4
x
2
Solution:
3
10
(a) y = ( 6 x + 3 x + 1 )
dy
9
3
3
= 10 ( 6 x + 3 x + 1 )  ( 6 x + 3 x + 1 )'
dx
9
2
3
= 10 ( 18 x + 3 ) ( 6 x + 3 x + 1 )
2
9
3
= 30 ( 6 x + 1 ) ( 6 x + 3 x + 1 )
(b) y = x 2 3x 1
dy 2x 3
=
dx 2 x 2 3x 1
 x 2  x 2 
(c) y =
x
x 2 4
=
x
 1
2
= ( x – 4 ) x 2
3 1
= x 2  4 x 2
dy 3 1  3
= x 2  2 x 2
dx 2
9
(d) y =
2
 x    1 4
x
– 4
2
= 9 ( x + x + 1 )
dy
– 5
2
2
= – 36 ( x + x + 1 ) ( x + x + 1 )
dx
– 5
2
= – 36 ( x + x + 1 ) ( 2 x + 1 )
 36 2 x 1 
=
 x 2  x 1  5
Calculus 17 Unit (1)