Basic Rules of Differentiation

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Basic Rules of Differentiation

By

Dr. Julia Arnold and Ms. Karen

Overman

using Tan’s 5th edition Applied

Calculus for the managerial , life,

and social sciences

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Basic Rules of Differentiation

Using the definition of the

derivative,

can be tiresome.

In this lesson we are going to learn

and use

some basic rules of differentiation

that are

derived from the definition. For

these rules,

let’s assume that we are discussing

differentiable functions.

 





 

h

x f h

x f x

f

h

 



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Rule 1:

Derivative of a constant

, for any constant,c.

This rule states that the derivative of a constant is zero.

For example,

 

0



dx

c

d

5

)

(

x



f

0

)

(

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 

 n1

n

nx dx

x d

Rule 2:

The Power Rule

, where n is any real number

This rule states that the derivative of x raised to a power is the power times x raised to a power one less or n-1.

For example,

 

5

x

f



 

4

5

'

x

f



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

( )





f (x)



dx

d c x

cf dx

d



Rule 3:

Derivative of a Constant Multiple of a Function

, where c is a constant

This rule states that the derivative of a constant times a function is the constant times the derivative of the function.

For example, find the derivative of f

 

x 5x4

 





 

3 3

4 4

20 4 5

5 5

x x

x dx

d x dx

d x

f dx

d

 

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Rule 4:

Derivative of a Sum or Difference



(

)

(

)





(

)





g

(

x

)



dx

d

x

f

dx

d

x

g

x

f

dx

d







This rule states that the derivative of a sum or difference is the sum or difference of the derivatives.

For example, find the derivative of x2_{+2x -3}





 

2 2

0 2 2

3 2

3

2 2

2

 

  

 

  

x x

dx d x

x dx

d

The derivative of x squared is done by the Power Rule (2), the

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More Examples:

Find

 

_x dx

d

You’ll notice none of the basic rules specifically mention

radicals, so you should convert the radical to its exponential form, X1/2 _{and then use the power rule.}

 

x

dx

d

x

dx

d

2

1

2

1

2

1

2

1

2 1 2

1 1

2 1 2

1

















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More Examples:

Find _

  

 

2 1 x dx

d

Again, you need to rewrite the expression so that you can use one of the basic rules for differentiation. If we rewrite the fraction as x-2 _{,then we can use the power rule.}

 

2 2 1 3 ₃

2

1

x

dx

d

x

dx

d























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More Examples:

Find         x x x dx

d 4 3 2 7

Rewrite the expression so that you can use the basic rules of differentiation. 1 2 3 3 7 2 4 7 2 4 7 2

4   _ _ _ _ _ _ 

x x x x x x x x x x

Now differentiate using the basic rules.

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Another example:

Find the slope and equation of the tangent line to the curve

y



2

x

2



1

at the point (1,3).

Recall from the previous chapter, the derivative gives the slope of the tangent to the curve. So we will need to find the

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Find the slope and equation of the tangent line to the curve at the point (1,3).

1

2





x

y

 





 

x

dx

d

x

dx

d

x

dx

d

y

dx

d

4

0

2

1

2

1

2

1 2

2 2



















First find the derivative of the function.

Now, evaluate the derivative at x = 1 to find the slope at (1,3).

4

1

4





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Example continued:

Now we have the slope of the tangent line at the point (1,3) and we can write the equation of the line.

Recall to write the equation of a line, start with the point slope form and use the slope, 4 and the point (1,3).









1

4

3

1

4

3

1 1



















x

y

x

y

x

y

x

m

y

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The graph below shows the curve in blue and the tangent line at the point (1,3), in red.

1

2





x

y

1

4



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Another example:

Find all the x values where has a horizontal tangent line.

x

y



3



2



Find the derivative.



x3 2x2  x



 3x2 4x1

dx d

Since horizontal lines have a slope of 0, set the derivative equal to 0 and solve for x.







1 or

3 1

0 1 or

0 1 3

0 1

1 3

0 1 4

3 2

  



  



 



  

x x

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Rule 5: The Product Rule



f

(

x

)

g

(

x

)



f

(

x

)

g

(

x

)

g

(

x

)

f

(

x

)

dx

d









In other words:

The derivative of f times g is

the first times the

derivative of the second plus the second times the

derivative of the first.

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Example of the Product Rule:

Find the derivative of f(x)  (2x2 1)(3x  4)

Since the function is written as a product of two functions you

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2. Same derivative by expanding and using the Power Rule. 3 16 18 ) ( 4 3 8 6 ) ( ) 4 3 )( 1 2 ( ) ( 2 2 3 2            x x x f x x x x f x x x f

1. The Product Rule

 





_               first the of derivative second second the of derivative first









3 16 18 ) ( 16 12 3 6 ) ( 4 ) 4 3 ( 3 ) 1 2 ( ) ( 1 2 ) 4 3 ( 4 3 ) 1 2 ( ) ( 2 2 2 2 2 2                          x x x f x x x x f x x x x f x dx d x x dx d x x f

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Notice in the first example, finding the derivative of the

function , the derivative could be found

two different ways. Whether you use the Product Rule or

rewrite the function by multiplying and find the derivative

using the Power Rule, the result or the derivative,

, was the same.

) 4 x 3 )( 1 x

2 ( ) x (

f  2  

 

18

16

3

'

x



x

2



x



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Example 2 of the Product Rule:

Find f’(x) for

  







 

3 2 1 2 1 3 2 1 3

1

)

(

first

the

of

derivative

second

the

of

derivative

first

'

Rule.

Product

the

use

Now

1

)

(

form.

l

exponentia

in

radical

the

rewrite

First

x

dx

d

x

dx

d

x

f

x

f

x

f



















































































 

x



x

3



x



1



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 

2 2 5 2 2 5 2 5 2 2 1 2 1 3 3 2 1 2 1 3 3 2 7 ) ( exponents. the add you base same the g multiplyin are you when Recall 3 3 2 1 ) ( 3 1 0 2 1 ) ( 1 1 ) ( x x x f x x x x f x x x x x f x dx d x x dx d x x f                                            

Example 2 continued…

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Rule 6: The Quotient Rule

This rule may look overwhelming with the functions but it is easy to learn if you can repeat these words: The derivative of a quotient is the bottom times the derivative of the top minus the top times the derivative of the bottom over the bottom squared.













2

) x ( g ) x ( g dx d ) x ( f ) x ( f dx d ) x ( g ) x ( g ) x ( f dx

d _ 

     





 





2

bottom

the

of

derivative

top

the

of

derivative

bottom

quotient

a

of

Derivative





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



 





2

bottom

the

of

derivative

top

the

of

derivative

bottom

quotient

a

of

Derivative













































Here it is written as a fraction:

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Example of the Quotient Rule:

Find         x 1 x 3 x dx d 2

First we will find the derivative by using The Quotient Rule





 





2 2 2 2 2 2 2 2 2 2 2 1 1 3 3 2 1 ) 1 3 ( 3 2 ) 1 3 ( 1 3 1 3 x x x x x x x x x x x x x x dx d x x x x dx d x x x x dx d                            





 





2

bottom bottom the of derivative top top the of derivative bottom quotient a of

Derivative _

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

1



 

2 2 ₂ 2 ₂ 1

2 2

1

0

1

3

.

derivative

the

find

Now

3

1

3

1

3

x

dx

d

x





























 



Another way to do the same problem is to do the division first and then use the power rule.

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Example 2: Find the derivative of

 

1

2





x

f

For this quotient doing the division first would require polynomial long division and is not going to eliminate the need to use the

Quotient Rule. So you will want to just use the Quotient Rule.

 





   













 







