Rules_for_Differentiation.ppt

The Power Rule
and other
Rules for Differentiation
Mr. Jayson Santelices

Taking the derivative by using the definition is a lot of work.
Perhaps there is an easy way to find the derivative.

Objective
 To differentiate functions using the
power rule, constant rule, constant
multiple rule, and sum and difference
rules.

The Derivative is …
 Used to find the “slope” of a function at a point.
 Used to find the “slope of the tangent line” to
the graph of a function at a point.
 Used to find the “instantaneous rate of change”
of a function at a point.
 Computed by finding the limit of the difference
quotient as ∆x approaches 0. (Limit Definition)

Notations for the
Derivative of a Function
d
dx
'( )
f x
'
y
dy
dx
dy
dx
“f prime of x”
“y prime”
“the derivative of y with respect to x”
is a verb. “Take the derivative with respect to x…”
is a noun.

 Differentiation is the process of
computing the derivative of a
function.
You may be asked to:
 Differentiate.
 Derive.
 Find the derivative of…

 Working with the definition of the
derivative is important because it
helps you really understand what the
derivative means.

The Power Rule
1
[ ] , is any real number
N N
d
x Nx N
dx


[ ] 1
d
x
dx


The Constant Rule
 The derivative of a constant function
is zero.
[ ] 0, is a constant
d
c c
dx


The Constant Multiple Rule
   
[ ( ) ] '( ) , is a constant
d
c f x c f x c
dx

 The derivative of a constant times a
function is equal to the constant
times the derivative of the function.

Constant Rule
 Find the derivative of:
( ) 7
f x 
'( ) 0
f x 
3
y  
0
dy
dx
 or ' 0
y 

The Sum and Difference Rules
[ ( ) ( )] '( ) '( )
d
f x g x f x g x
dx
  
[ ( ) ( )] '( ) '( )
d
f x g x f x g x
dx
  
The derivative of a sum is the sum of the derivatives.
The derivative of a difference is the difference of the derivatives.

Power Rule
 Differentiate:
3
( )
f x x

2
'( ) 3
f x x

9
y x

8
9
dy
x
dx

100
( )
g x x

99
'( ) 100
g x x


Constant Multiple Rule
1
3
2
y x

 
2
dy
dx

2
3
1
3 x

2
3
2
3
dy
dx x


2
4
( )
5
x
f x 
 
4
5
'( )
f x  2x
8
'( )
5
f x x

2
4
5
x


7
( ) 5
g x x

6
'( ) 35
g x x


Rewriting Before Differentiating
Function Rewrite Differentiate Simplify
3
5
( )
2
f x
x
 3
5
( )
2
f x x
 4
5
'( ) ( 3 )
2
f x x
  4
15
'( )
2
f x
x
 

2
7
( )
3
g x
x
 2
7
( )
3
g x x

7
'( ) (2 )
3
g x x

14
'( )
3
g x x


( )
h x x

1
2
( )
h x x

1
2
1
'( )
2
h x x

 1
2
1
'( )
2
h x
x


2
3
1
( )
2
j x
x

2
3
1
( )
2
j x
x

5
3
1 2
'( )
2 3
j x x

 
 
 
 
5
3
1
'( )
3
j x
x
 
2
3
1
( )
2
j x x



Sum & Difference Rules
 Differentiate:
2
( ) 5 7 6
f x x x
  
'( )
f x 
6 5 2
( ) 4 3 10 5 16
g x x x x x
    
'( )
g x 
10x 7

5
24x 4
15x
 20x
 5


Conclusion
 Notations for the derivative:
 The derivative of a constant is zero.
 To find the derivative of f (x) = xN
1. Pull a copy of the exponent out in
front of the term.
2. Subtract one from the exponent.
'( )
f x '
y
dy
dx

Rules_for_Differentiation.ppt

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Rules_for_Differentiation.ppt