Basic Rules of Differentiation

  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

  

Basic Rules of Differentiation

Using the definition of the f x  h  f x

      derivative,

f ' x  lim

    _{h 0} h 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

  Derivative of a constant

Rule 1:

  d c  

   , for any constant,c. dx This rule states that the derivative of a constant is zero.

  For example,

  f x ( ) 

  5 f x

  ' ( )  The Power Rule

  Rule 2: _n d x

    n  ¹ nx

   dx , 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 f x  x

   

  4 f ' x  5 x

    Derivative of a Constant Multiple of a Function

Rule 3:

  d d cf x  c f x ( ) ( )

     

  , where c is a constant

  dx dx

  This rule states that the derivative of a constant times a function is the constant times the derivative of the function. ₄

  f x  5x  

  For example, find the derivative of

  d d

₄

f x  5 x

        dx dx d

₄

x

  

  5

 

dx ₃ x

  

  5

  4   ₃ x

  

  20 Derivative of a Sum or Difference

Rule 4:

        ) ( ) ( ) ( ) ( x g 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 x

  2

• 2x -3

  2

         

  2

  2

  3

  2

  3

  2 ^{2 2}           x x dx d x dx d x dx d x x dx d

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

  2

  Find

More Examples:

  You’ll notice none of the basic rules specifically mention radicals, so you should convert the radical to its exponential form, X

  1

   

       

  1     

  2

  1

  2

  1

  1

  2

  2

  1/2 and then use the power rule.

    x dx d

  2

  

1

  

2

  1

  2

  1

  2

    x x x x x dx d x dx d

  1 Find

More Examples:

     

    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.

  3

  3

  1

   

  2

  2

  2

  2

  2

  1 x x x x dx d x dx d

            

   

   

  2 Find

More Examples:

  7

  7

  2

  4

  7

  2

  4

  2

  2

  4

  x x x dx d dx d x dx d x x dx d x x x dx d

          

       

    

   

     

  4

    



  





  x x x dx d

  7

  7

  2

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

  7

  2

  4

  2

  1

  4

  7

  2

  4 

  

     

  x x x x x x x x x x Now differentiate using the basic rules.

       

    _{1 1} ^{1 2 1 2 3}

  7 Find the slope and equation of the tangent

  Another example: ₂ line to the curve y  x  at the point (1,3).

  2

  of the tangent to the curve. So we will need to find the derivative and evaluate it at x = 1 to find the slope at the point (1,3). Then we’ll use the slope and the point to write the equation of the tangent line using the point slope form.

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

  1

  2 ²   x y  

        x

x

dx d

x

dx d x dx d y dx d

  4

  2

  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    m

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

Example continued:

  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

  4

  4

  3

  1

  4

  3 ^{1 1}        

     x y x y x y x x m y y

  2 y  x 2 

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

  4

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

   x x x y   ^{2 3}

Another example:

   

  1

  Thus the x values where the function has horizontal tangents is at x = -1 , -1/3.

  x x x x x x x x

    

        

   

  3 ²  

  4

  1

  3

  1

  1

  2 Find the derivative.

  1

  1 1 or

  1 or

    

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

  2 ^{2 2 3}      x x x x x dx d

  3

  4

  3 The graph below shows the function from the last example and the horizontal tangent lines at x=-1 and x=-1/3.

  Rule 5: The Product Rule d  

    f ( x ) g ( x ) f ( x ) g ( x ) g ( x ) f ( x )   dx

  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.

  It is helpful to learning the rule if you are able to repeat the words in red as you are using the rule.

Example of the Product Rule:

  2    f ( x ) (

  2 x 1 )( 3 x 4 )

  Find the derivative of Since the function is written as a product of two functions you could use the product rule or multiply it out and use the basic rules of differentiation. We’ll look at it BOTH ways.

  2    f ( x ) ( 2 x 1 )( 3 x 4 )

  Find the derivative of .

  1. The Product Rule

  

derivative of derivative of

   

first   second 

     

   

the second the first

   

₂ d d ₂ f  x x x x x

  ( )  ( 2  1 )  3  4  ( 3  4 )  2 

  1     ₂ dx dx f  x x x x

  ( )  ( 2  ₂ 1 )  3  ( ₂ 3  4 ) 

  4 f  x x x x

  ( )  6  ₂ 3  12 

  16  f x x x

  ( )  18  16 

  3 2. Same derivative by expanding and using the Power Rule. ₂ f x  x  x 

  ( ) (

  2 ₃ 1 )( ₂

  3 4 ) f x x x x

  ( )  6  8  3 

  4 Notice in the first example, finding the derivative of the ₂

     f ( x ) ( 2 x 1 )( 3 x 4 )

  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,

  2 , was the same. f x x x

  '  18  16 

  3  

  Example 2 of the Product Rule: ₃ f x x x

   

  1  

  Find f’(x) for   First rewrite the radical in exponentia l form. _{1 3}   ₂

  f ( x )  x x 

  1     Now use the Product Rule. derivative of derivative of     f ' x  first   second 

            the second the first     _{1 1}     ₃ d d _{2 2 3}

   f ( x )  x x  1  x  1 x

        dx dx

      Example 2 continued… _{1 1}     ₃ d d _{2 2 3}

  

  f ( x ) x x

  1 x 1 x    

   

     

  dx dx

     

   _{1 1}

      ₃

  1 _{2 2 2}

  f  ( x ) x x x

  1 3 x      

     

  2     _{5 5}

  1 _{2 2 2} 

  f ( x )  x 

  3 x  3 x Recall when you are multiplyin g

  2 ₅ the same base you add the exponents.

  7 _{2 2} 

  f ( x ) x

  3 x  

  2

• The derivative could also be found by performing the

  Rule 6: The Quotient Rule

  d d  g ( x )  f ( x )  f ( x )  g ( x ) 

    d f ( x ) dx dx

   ₂   dx g ( x ) g ( x )

   

    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. derivative of derivative of     bottom   top 

          Derivative of

    the top the bottom    

Here it is written as a fraction:

  derivative of derivative of     bottom   top 

          Derivative of

    the top the bottom      ₂   a quotient bottom

     

  Keep in mind that when we say “top” and “bottom” we are referring to the numerator and denominator of the original function respectively. Another important thing to remember with this rule is the order in the numerator of the derivative!!

  Find

  3 x dx d ² First we will find the derivative by using The Quotient Rule

Example of the Quotient Rule:

      x 1 x

  3 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

    

     

       

      

  Derivative of    

        ² bottom bottom the of derivative top

top the

of derivative bottom quotient a

     

           

             

  1

       

     

  1

  

2

) 1 3 (

  3

  2 1 ) 1 3 (

  3

  3

  1

  2 ^{2 2 2 2 2 2 2 2}

  3

      ₂ ² ₂ ²

²

^{1 1 2 2}

  1

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

     

                      

  3 x x x x x x x dx d x x x x x x x x x x

  1

  3

  3

  1

  3 . derivative the find Now

  1

  1

  1

  1

  1

  Again, notice there is more than one method you could use to find the derivative.

  x f x Example 2: Find the derivative of 

    ₂ x 

  1 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. ₂

d d

₂ x 

  1  x  x  x 

  1

   

  

   

dx dx

f ' x 

    _{2 2} x 

  1 ₂   _{2 2} x  1  1  x  2 x x  1  2 x

     

    _{2 2 2 2} x  1 x 

  1 ₂      

  1 x