The document discusses rules for differentiation, including the power rule, constant rule, constant multiple rule, and sum and difference rules. It explains that the power rule states that the derivative of x^N is Nx^(N-1), the constant rule is that the derivative of a constant is 0, and the constant multiple rule is that the derivative of c*f(x) is c*f'(x). It also explains that the sum and difference rules state that the derivative of a sum is the sum of the derivatives, and the derivative of a difference is the difference of the derivatives. Several examples are worked out applying these rules to find derivatives of various functions.

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

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

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

4. 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)

5. 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.

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

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

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

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

18. 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.

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

26. 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.

37. 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


38. Constant Multiple Rule
 Find the derivative of:
1
3
2
y x

 
2
dy
dx

2
3
1
3 x

2
3
2
3
dy
dx x


39. Constant Multiple Rule
 Find the derivative of:
2
4
( )
5
x
f x 
 
4
5
'( )
f x  2x
8
'( )
5
f x x

2
4
5
x


40. Constant Multiple Rule
 Find the derivative of:
7
( ) 5
g x x

6
'( ) 35
g x x


41. 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
 

42. Rewriting Before Differentiating
Function Rewrite Differentiate Simplify
2
7
( )
3
g x
x
 2
7
( )
3
g x x

7
'( ) (2 )
3
g x x

14
'( )
3
g x x


43. Rewriting Before Differentiating
Function Rewrite Differentiate Simplify
( )
h x x

1
2
( )
h x x

1
2
1
'( )
2
h x x

 1
2
1
'( )
2
h x
x


44. Rewriting Before Differentiating
Function Rewrite Differentiate Simplify
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



45. 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


46. 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