# Rules for Differentiation

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We will discuss some basic rules for differentiation.
1. 3.3 Rules for Differentiation
Colorado National Monument
Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington
2. If the derivative of a function is its slope, then for a
constant function, the derivative must be zero.
d example: y 3
 c  0
dx y 0
The derivative of a constant is zero.

3. If we find derivatives with the difference quotient:
2
d 2
x lim
 x  h  x
lim
2
 x 2
 2 xh  h 2
  x 2
2x
dx h 0 h h 0 h 1
1 1
3
d 3  x  h   x3 1 2 1
x lim 1 3 3 1
dx h 0 h 1 4 6 4 1
2
lim
 x 3
 3 x 2
h  3 xh 2
 h  3
 x 3
3x 2
1 5 10 10 5 1
h 0 h (Pascal’s Triangle)
2 3
d 4
x lim
 x4
 4 x h  3
6 x h  2
4 xh2
 h   x34 4
4x3
dx h 0 h
We observe a pattern: 2x 3x 2 4x 3 5x 4 6x 5 …

4. We observe a pattern: 2x 3x 2 4x 3 5x 4 6x 5 …
examples:
d n
dx
 x  nx n 1
f  x  x 4 y  x8
f  x  4 x 3 y 8 x 7
power rule

5. constant multiple rule:
examples:
d du
 cu  c d n
cx cnx n  1
dx dx
dx
d
7 x5 7 5 x 4 35 x 4
dx
When we used the difference quotient, we observed that
since the limit had no effect on a constant coefficient,
that the constant could be factored to the outside.

6. constant multiple rule:
d du
 cu  c
dx dx
sum and difference rules:
d du dv d du dv
 u  v    u  v  
dx dx dx dx dx dx
4 2
4
y  x  12 x y x  2 x  2
(Each term
dyis treated separately)
3 3
y 4 x  12 4 x  4 x
dx 
7. Find the horizontal tangents of: y x 4  2 x 2  2
dy
4 x 3  4 x
dx
Horizontal tangents occur when slope = zero.
4 x 3  4 x 0 Plugging the x values into the
original equation, we get:
x 3  x 0
y 2, y 1, y 1
x  x  1 0
2
(The function is even, so we
x  x  1  x  1 0 only get two horizontal
tangents.)
x 0,  1, 1

8.
9. y x 4  2 x 2  2
10. y x 4  2 x 2  2
y 2
11. y x 4  2 x 2  2
y 2
y 1
12. y x 4  2 x 2  2
13. y x 4  2 x 2  2
dy
4 x3  4 x
dx
First derivative
(slope) is zero at:
x 0,  1, 1

14. product rule:
d dv du Notice that this is not just the
 uv  u  v
dx dx dx product of two derivatives.
This is sometimes memorized as: d  uv  u dv  v du
d  2 3
dx 
x  3 
2 x  5 x  
  x 2  3  6 x 2  5    2 x 3  5 x   2x 
 
2 x 5  5 x3  6 x3  15 x
 
d 6 x 4  5 x 2  18 x 2  15  4 x 4  10 x 2
dx

2 x 5  11x 3 15 x 
10 x 4  33x 2  15 10 x 4  33 x 2  15

15. quotient rule:
du dv
v u  u  v du  u dv
d u dx dx or d  
   2
dx  v  v2 v
  v
3
d 2 x  5x

    
x 2  3 6 x 2  5  2 x3  5 x  2 x 
2
dx x  3 2
 x  3
2

16. Higher Order Derivatives:
dy
y  is the first derivative of y with respect to x.
dx
dy d dy d 2 y is the second derivative.
y    2
dx dx dx dx (y double prime)
dy 
y  is the third derivative. We will learn
dx later what these
higher order
d derivatives are
 4
y  y is the fourth derivative. used for.
dx