Chapter 3 Section 3
 Prerequisites for Calculus

Integer Powers, Multiples, Sums, and Differences:
Each Rule is a different color:

d (c)=0 d (xn)=nxn-1 d (x)=1    d (cu)=c du
dx dx dx dx dx
d

(-u)=

 -1 d (u)=

-

 du d (cxn)=cnxn-1
dx dx dx dx

Horizontal Tangents occur when:

dy = 0
dx

Example: Does the curve y = x4 - 8x2 + 6 have any
                horizontal tangents.

dy = 4x3-16x
dx

set it = 0 ® 4x3-16x = 4x(x2-4)=0  ®  x = 0, 2, -2
Points: (0,6)  (2,-10)  (-2,-10)

 

Example Horizonatal Tangents if any for:
f(x)= x4 - 2x3 + x2 - 4x + 9

dy = 4x3 - 6x2 - 2x -4
dx

No rational zeros. Using solve on the calculator we find that  when f '(x) = 0, x = 2.127
 

Product Rule

d (uv)  =  u dv + v du
dx dx dx

Example: Find y ' for :
y = (3x+5)(x2 -6)
       (3x+5)(2x) + (x2 -6)(3)
        6x2+10x+3x2-18 
y ' = 9x2 -8

 

Quotient Rule

Example: Find f ' (x) for :
f(x) = x2 - 5   =    2x+8(2x) - x2 - 5(2)
         2x + 8                  (2x + 8)2  
                            4x2 +16x - x2 -10
                                    (2x + 8)2  
                    f(x) ' =   3x2 +16x-10
                                    (2x + 8)2  

 

Negative Integer Powers of X

d (xn) = nxn-1
dx

Examples:

1) d ( 4 ) -4x-2 = -4 2) d (2x+ 5 )    = 2- 10
   dx x x2    dx x2 x3