The Linear Property and the Power Rule and the shift rule
Reminders:
As seen before the derivative of f ( x )  x 2 is
The derivative of a line is the slope of the line.
f ' ( x)  2 x .
The linear property says (af+bg)' (x)= af '(x) + bg'(x). That allows us to take the
derivative of any quadratic.
f ( x )  3x 2  15 x  12
f ' ( x )  3(2 x )  15  6 x  15
d n
Power rule: For n  0,
( x )  nxn 1
ie for f ( x )  x n ,
dx
d 3
d 10
( x )  3x 2
( x )  10 x 9
dx
dx
Example:
Examples:
1
2
1
d
1 
(x )  x 2
dx
2
f ' ( x )  nxn 1
d 1
d)
1
( )  ( x 1 )   x  2 
dx x
dx
x2
The linear rule applied to polynomials:
f ( x )  2 x 4  8 x 3  7 x 2  3x  2
f ' ( x )  2( 4 x 3 )  8(3x 2 )  7( 2 x )  3  8 x 3  24 x 2  14 x  3
Examples:
d
(12 x 5  4 x 3  13 x  8)  12(5 x 4 )  4(3x 2 )  13  60 x 4  12 x 2  13
dx
Examples: Fractional and negative exponents:
1
f ( x)  3 x  x 3
2
1 
f ' ( x)  x 3
3
and
d 1
d
2
( )  ( x  2 )  2 x  3 
2
dx x
dx
x3
Applying the linear rule:
2
d 3
5
4 
10
(4 x  )  x 3 
2
dx
3
x
x3
When the function is a quotient and the denominator is a power of x:
d x2  7x  9
d x2 7x 9
d
(
) (

 )  ( x  7  9 x 1 )
x
dx x
x x
dx
Example: dx
9
 1  9x 2  1 
x2
Example:
d x2 1
d
3
1
(
)  ( x 3 2  x 1 2 )  x 1 2  x  3 2
dx
dx
2
2
x
One rule, I call the shift rule, you will not find in the book until we get to the chain rule.
But it saves work and allows us to do more examples before we do the chain rule.
Suppose you shift f(x) to the right 2 units. You don't change the shape of the graph. The
tangent line should shift right 2 units also, but have the same slope.
The shift rule says
d
( f ( x  a ))  f ' ( x  a )
dx
Examples:
f ( x )  ( x  2) 3
f ' ( x )  3( x  2) 2
1
g( x)  x  1

1
g ' ( x )  ( x  1) 2
2
This only works for a shift. It does not work for any other composition.