Page 23 - (1-1)eng

(1 - 1) The First Derivative of a Function

Summary

1) Constant function y = c , c   y' = 0

2) Power rule y = x , n   y' = n x n – 1 y  1  dy   1
n
x dx x 2
y  x  dy  1
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 y  ƒ x y' = ƒ' g  2 g ƒ '
 
g x g

7) Chain rule y= ƒ ( z) and z = g ( x ) dy dy dz
 
dx dz dx
 g x
y = ( ƒ  g ) ( x ) y = ƒ  
 g x
y'( x ) = ƒ'   × g' ( x )

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

9) Parametric function First derivative
If x , y are each expressed in terms of dx dy
(a) Find and separately.
dt dt
a third variable t say, called the parameter,

dy dy dt
then x = ƒ( t ) , y = g ( t ) give the (b) Use  
dx dt dx
parametric form of the equation relating x

and y.

Calculus 22 Unit (1)