Common Derivatives and Integrals 
                       
                      You can navigate to specific sections of this handout by clicking the links below. 
                       
                      Derivative Rules: pg. 1 
                      Integral Formulas: pg. 3 
                      Derivatives Rules for Trigonometric Functions: pg. 4 
                      Integrals of Trigonometric Functions: pg. 5 
                      Special Differentiation Rules: pg. 6 
                      Special Integration Formulas: pg. 7 
                       
                      Derivative Rules: 
                       
                                                d [   ]      ′
                      1. Constant Multiple Rule dx cu = cu , where c is a constant.   
                       
                                                  d [      ]    ′    ′
                      2. Sum and Difference Rule  dx u ± v = u ± v  
                       
                                       d [   ]     ′      ′
                      3. Product Rule  dx uv = uv + vu  
                       
                      4. Quotient Rule  d u = vu′−uv′  
                                       dx v         v2
                                            
                       
                                         d [ ]
                      5. Constant Rule,  dx c = 0  
                       
                                     d [ n]=      n−1 ′
                      6. Power Rule  dx u      nu    u  
                       
                                     d [ ]
                      7. Power Rule  dx x =1 
                       
                      8. Derivative Involving Absolute Value  d [  ]   u ( ′)         
                                                             dx u = u u ,u ≠ 0
                       
                                                                        d [    ]   u′
                      9. Derivative of the Natural Logarithmic Function  dx lnu = u  
                                                                                    Common Derivatives and Integrals 
                      Provided by the Academic Center for Excellence      1                          Reviewed June 2008 
                                        
                       
                                                                     d [ u]= u ′
                      10. Derivative of Natural Exponential Function  dx e     e u  
                       
                       
                                                               ( )   (        2 )(      2 )
                      Example 1:  Find the derivative of  f x = 4x −3x           3+2x  
                       
                      Since there are two polynomials multiplied by each other, apply the third derivative rule, the 
                      Product Rule, to the problem.   
                       
                      This is the result of the Product Rule: 
                       
                        ′               2   d [       2 ]          2  d [         2 ]  
                         ( )   (          )                (        )
                       f  x = 4x−3x dx 3+2x + 3+2x dx 4x−3x
                       
                      Now, take the derivative of each term inside of the brackets. Multiple derivative rules are 
                      used, including the Sum and Difference Rule, Constant Rule, Constant Multiple Rule, and Power Rule. 
                      When applied, the result is: 
                       
                        ′               2                    2
                         ( )              (       )            (      )
                               (         )           (        )
                      f   x = 4x−3x 0+4x + 3+2x 4−6x  
                       
                      Simplify: 
                       
                        ′               2                2
                         ( )              (   )            (       )
                               (         )       (        )
                      f   x = 4x−3x 4x + 3+2x 4−6x  
                       
                      Multiply the polynomials by each other:  
                       
                        ′           2       3            2              3
                         ( )   (             )  (                        )
                      f   x = 16x −12x + 12+8x −18x−12x  
                       
                        ′          2       3           2             3
                      f (x)=16x −12x +12+8x −18x−12x  
                       
                      Combine like terms to get a simplified answer: 
                       
                        ′            3       2
                      f (x)= −24x +24x −18x+12 
                       
                       
                       
                       
                       
                       
                                                                                   Common Derivatives and Integrals 
                      Provided by the Academic Center for Excellence      2                        Reviewed June 2008 
                                         
                      Integral Formulas: 
                       
                      Indefinite integrals have +C as an arbitrary constant.  
                       
                            ( )            ( )
                      1.∫kf u du = k∫ f u du, where k is a constant. 
                       
                             ( )     ( )          ( )         ( )
                          [             ]
                      2.  ∫ f u ± g u du = ∫ f u du ± ∫ g u du 
                       
                      3.  ∫ du = u + C  
                       
                                     n+
                                    u 1
                      4.  ∫undu =        +C,n≠−1 
                                   n+1
                       
                      5.  ∫ du = lnu +C 
                           u
                       
                      6.  ∫eudu = eu +C  
                       
                                                (   2      3     )
                      Example 2: Evaluate ∫ 4x −5x +12 dx 
                       
                      To evaluate this problem, use the first four Integral Formulas. First, use formula 2 to make 
                      the large integral into three smaller integrals: 
                       
                        (   2     3      )          2          3
                      ∫ 4x −5x +12dx=∫4x dx−∫5x dx+∫12dx 
                       
                      Second, pull out the constants by using formula 1: 
                       
                      =4∫x2dx−5∫x3dx+12∫dx 
                       
                      Now find each integral using formulas 3 and 4: 
                       
                          x2+1      x3+1 
                                               ( )
                      =4          −5          +12 x +C 
                          2 +1     3+1
                                         
                       
                       
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                                                                                                                 Integrals 
                                        
                     Although three integrals have been removed, only one constant C is needed because C 
                     represents all unknown constants. Therefore multiple C’s can be combined into just one C. 
                     To get the final answer, simplify the expression: 
                      
                          x3      x4                
                                     
                      =4        −5        +12x+C
                          3       4 
                                     
                      
                     =4x3 −5x4 +12x+C 
                        3      4
                      
                      
                     Derivatives Rules for Trigonometric Functions:  
                      
                        d      ( )    (   ( )) ′
                           [      ]
                     1. dx sin u   = cos u u  
                      
                         d      ( )     (   ( )) ′
                            [      ]
                     2.  dx cos u    =−sin u u  
                      
                         d      ( )        2( ) ′
                            [      ]   (        )
                     3.  dx tan u   = sec u u  
                      
                         d      ( )          2( ) ′
                            [      ]    (        )
                     4.  dx cot u   =−csc u u  
                      
                         d      ( )    (   ( )    ( )) ′
                            [      ]
                     5.  dx sec u   = sec u tan u u  
                      
                         d      ( )     (   ( )    ( )) ′
                            [      ]
                     6.  dx csc u   =−cscu cot u u  
                                                                       ( )
                     Example 3: Find the derivative of  f (x)= sin x  
                                                                    cos(x)
                      
                      
                     When finding the derivatives of trigonometric functions, non-trigonometric derivative rules 
                     are often incorporated, as well as trigonometric derivative rules. Looking at this function, 
                     one can see that the function is a quotient. Therefore, use derivative rule 4 on page 1, the 
                     Quotient Rule, to start this problem:  
                     Provided by the Academic Center for Excellence            4             Common Derivatives and 
                                                                                                               Integrals