Download Derivative Rules

Derivative Rules
3.3
Derivative of a Constant
Derivative of a Constant
For any constant C, d (C )  0
dx
Suppose f(x) is a constant function, that is, f(x) = C,
where C is a constant. Since the value of the function
never changes, the instantaneous rate of change must
be zero.
Examples:
d
(2)  0
dx
f '( )  0
dy
1
d 1
if y      0
dx
2
dx  2 
Derivatives of Powers
Derivative of a Power
If n is any real number (n may or may not be an integer),
d n
( x )  nx n1
dx
Another way to remember this rule is “power in front,
reduce the power by one”.
Examples:
d 4
( x )  4 x41  4 x3
dx
f '(v
5 3
5  5 31
) v
3
5 8 3
 v
3
dy
1
dy d  1  d 2
2
if y  2  t 
  2    t   2t 21  2t 3
dt
t
dt dt  t  dt
constant multiple rule:
d
du
 cu   c
dx
dx
examples:
d n
cx  cnx n 1
dx
d
7 x 5  7  5 x 4  35 x 4
dx

constant multiple rule:
d
du
 cu   c
dx
dx
sum and difference rules:
d
du dv
u  v   
dx
dx dx
d
du dv
u  v   
dx
dx dx
y  x  2x  2
dy
3 separately)
(Each term is 
treated
3
4x  4x
y  4 x  12
dx
y  x  12 x
4
4
2

Derivatives of trig functions
d
sin x  cos x
dx
d
csc x   csc x cot x
dx
d
cos x   sin x
dx
d
sec x  sec x tan x
dx
d
tan x  sec 2 x
dx
d
2
cot x   csc x
dx
Example: Find the derivative of:
4 2
y  x  x 2
3
3
4
y '  3x   2 x  0
3
2
8
y '  3x  x
3
2
Example: Find the derivative of:
3
y 2
x
y  3x
y '  6 x
3
6
 3
x
2
Example: Find the derivative of:
y2 x
y  2x
1
y '  2  x
2

1
2
1
x


x
x
1
2
Example: Find the derivative of:
y  5  sin x
dy
 0  cos x
dx
dy
 cos x
dx
Example:
Find the horizontal tangents of:
y  x4  2x2  2
dy
 4 x3  4 x
dx
Horizontal tangents occur when slope = zero.
4 x3  4 x  0
x3  x  0
x  x  1  0
2
x  x  1 x 1  0
x  0, 1, 1
Plugging the x values into the
original equation, we get:
y  2, y  1, y  1
(The function is even, so we
only get two horizontal
tangents.)

Higher Order Derivatives
y' '
y' ' '
1st derivative
y'
d
f (x)
dx
dy
dx
f ' ( x)
f ' ' ( x)
f ' ' ' ( x)
3rd derivative
2nd derivative
d  dy  d y
  2
dx  dx  dx
2
3
d y
dx 3
Find the first 4 derivatives of:
y  x  3x  2
3
2
y '  3x  6 x
2
y' '  6 x  6
y' ' '  6
f ( x)  0
4
Find the derivative of:
y  ( x  1) 2
y  x2  2x 1
y'  2 x  2
We
have to
foil first!
Find the slope of the curve y = 2cos(x) at:

x
2
x

y  2 cos x
y '  2 sin x
3
x 
 
y' 
 2


2
  2 sin
2

  2 3
  
y' 

 3
  2 sin
3
2
 3 
y'    2 sin   0