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									  Section 4.4
The Chain Rule
F  x   f  g x 

F'  x   f '  g  x   g'  x 
     F  x   f  g x 
    F'  x   f '  g  x   g'  x 

        F  x    x  ln x 
                                4


                          3        1
F'  x   4  x  ln x   1       
                                   x
Try these two…

                         
Find f ‘ (x) if f  x   x  4x         
                                 2           3




                             
               f '  x   3 x  4x            2x  4
                                                 2
                                     2




                                         2
                           2x  3 
Find f ‘ (x) if f  x           
                           5x  2 
                           2x  3   2  5x  2   5  2x  3  
                f 'x  2                                      
                           5x  2           5x  2            
                                                          2
                                                                  
                                2x  3 
                f '  x   22              
                                 5x  2  
                                           3
                                            
                                  NO CALCULATOR

          
If h  x   x  4             1 then h'  2  
                         3/4
              2
                                  ,
A) 3              B) 2              C) 1             D) 0   E) DNE

                                    3 2
                                              
                                              1/4
                         h'  x     x 4          2x 
                                    4
                                    3
                         h'  2    0   4 
                                         1/4

                                    4
                               NO CALCULATOR
If y  cos2 x  sin2 x, then y ' 
A)  1 B) 0     C)  2  cos x  sin x  D) 2 cos x  sin x  E)  4cos x sin x

                            y  cos2x
                            y '   sin2x  2 
                            y '  4 sin x cos x
                  d5 y
If y  ekx , then 5 
                  dx
A) k 5 ek      B) k 5 ekx      C) 5!ekx           D) 5!e x   E) 5ekx

               y  ekx  y '  kekx  y "  k 2ekx etc.
                           NO CALCULATOR
 What is the instantaneous rate of change at x = 0 of the function
 f given by f  x   e2x  3 sin x
  A)  2      B)  1         C) 0                    D) 4       E) 5
                       f '  x   2e 2x  3 cos x
                       f '  0   2e             3cos  0 
                                        2 0 




The y-intercept of the tangent line to the curve y  x  3 at 1 2 is
                                                                ,
          1            1          3            5          7
      A)            B)         C)           D)         E)
          4            2          4            4          4
                                           1
        dy 1
                x  3
                         1/ 2
                                   y  2   x  1
        dx 2                               4
                                           1
                                   y  2   0  1
     dy          1
         |x 1 
     dx          4                         4
                                                 NO CALCULATOR

                 
If g  x   tan2 e x , g'  x  
A) 2e tan  e  sec  e 
         x                   x       2       x



B) 2 tan  e  sec  e 
                     x           2       x



C) 2 tan  e  sec  e 
             2           x               x



D) e sec  e 
     x           2           x
                                                                             e 
                                                   g'  x   2 tan e x sec 2 e x   x



E) 2e tan  e 
         x                   x
                                       NO CALCULATOR
If h  x    f  x    f  x  g  x  , f '  x   g  x  , and g'  x   f  x , then h'  x  
                       2
                      
A) f  x  g  x 
B) 2f  x   f  x  g  x 

C)  f  x   g  x  
                           2
                      
D)  f  x   g  x  
                           2
                      
E) g  x    2f  x  g  x    f  x  
              2                                  2
                                          

            h  x   f  x   f  x  g  x 
                                2
                             
          h'  x   2 f  x  f '  x   f '  x  g  x   g'  x  f  x 
                              
          h'  x   2  f  x  g  x   g  x  g  x   f  x  f  x 
                               
                     CALCULATOR REQUIRED
Let the function f be differentiable on the interval [0, 2.5] and
Use the table to estimate g ‘ (1) if g  x   f  f  x 
                                                         
              x      0      0.5         1        1.5        2    2.5
            f(x)    1.7     1.8         2        2.4       3.1   4.4
   A) 0.8          B) 1.2          C) 1.6              D) 2.0      E) 2.4
                             g  x   f f  x 
                                                
                            g'  x   f ' f  x  f '  x 
                                                  
                            g' 1  f '  f 1 f ' 1
                                               
                            g' 1  f '  2 f ' 1
                            g' 1   2  0.6 
                    NO CALCULATOR
d
dx
      
   lne3x 

                                        1          3
A) 1         B) 3      C) 3x      D)         E)
                                       e3x        e3x
                      y  3xlne
                         NO CALCULATOR
                             t2
The formula x  t   ln t      1 gives the position of an object
                             18
moving along the x-axis during the time interval 1  t  5. At
the instant when the acceleration of the object is zero, the
velocity is
                     1        2
A) 0              B)       C)           D) 1        E) undefined
                     3        3
                 t2                                  1 1
x  t   ln t     1
                                 1 1
                          v t   t          at  2 
                 18              t 9                  t   9
                                1 1                  1 1
                         v 3   3           0 2 
                                3 9                   t   9
                                                  t  3  t  3
                        NO CALCULATOR
                                                               
The slope of the line tangent to the graph of y  ln x at e2 , 1 is
   e2          2             1             1        1
A)          B) 2          C) 2         D)        E)
   2          e             2e            2e        e

                                 1
                            y  ln x
                                 2
                        dy        1
                               
                        dx 2x
                   dy             1
                      | e2 , 1  2
                   dx   2e
                               NO CALCULATOR
If f  x   ln  cos 2x  , then f '  x  
A)  2 tan2x        B) cot 2x        C) tan2x     D)  2cot 2x   E) 2 tan2x

                          f 'x 
                                       1      sin2x   2 
                                     cos 2x
                                   2 sin 2x
                          f 'x 
                                    cos 2x
                              NO CALCULATOR
If f  x   sin2x  ln  x  1 , then f '  0  
A)  1          B) 0             C) 1            D) 2         E) 3
                                                         1
                          f '  x   cos 2x  2  
                                                       x 1
                                                         1
                          f '  0   cos  0  2  
                                                       0 1
                                 NO CALCULATOR
                 2x  1, then g'  x  
       g x 
If e
     1                     2
A)                    B)                 C) 2  2x  1           D) e 2x 1 E) ln  2x  1
   2x  1                2x  1

                                                  ln  2x  1
                                        g x 
                                 ln e
                                  g  x   ln  2x  1
                                                1
                                  g'  x           2
                                             2x  1
                     NO CALCULATOR
A particle moves on the x-axis in such a way that its position
at time t, t  0, is given by x  t   ln x  . At what value of t does
                                                           2


the velocity of the particle attain its maximum?
A) 1                B) e1/ 2           C) e             D) e3 / 2    E) e2
                                                 1  2ln x
x  t   ln x               v  t   2ln x   
                    2

                                                x         x
                                         2
                                          x   x   12ln x 
                               at   
                                                     x2
                                         2  2ln x
                                    0
                                              x2
                                ln x  1
                            NO CALCULATOR
                                        d2 y
If y  ln  cos x  and 0  x  , what is 2 in terms of x?
                               2         dx
A) tan x       B)  tan x C) sec 2 x D)  sec 2 x E)  csc 2 x
                   dy   1
                             sin x    tan x
                   dx cos x
                  d2 y
                  dx 2
                          
                         sec 2 x   
                             NO CALCULATOR
                       
If f  x   ln x 2  e2x , then f ' 1 
A) 0             B) 1          C) 2              D) e         E) undefined

                        f 'x 
                                       1
                                   x 2  e2x
                                             
                                             2x  2e 2x   
                                   2  2e2
                         f ' 1 
                                    1  e2
                                NO CALCULATOR
If f  x   e2x and g  x   ln x, then the derivative of y  f  g  x   at x  e is
A) e2          B) 2e2           C) 2e           D) 2              E) undefined

                    y  f  g  x    y '  f '  g  x   g'  x 
                                                              1
                      f '  x   2e   2x
                                                   g'  x  
                                                              x
                 f '  g  x    2e2ln x

                                                      2e2lne
                           f '  g  e   g'  e  
                                                        e
                             CALCULATOR REQUIRED
The position of a particle moving on the x-axis, starting at t = 0,
is given by x  t    t  a   t  b  where 0  a  b
                               3


Which of the following statements is true?
I. The particle is at a positive position on the x-axis at time
     t = (a + b)/2 NO
II. The particle is at rest at time t = a YES
III. The particle is moving to the right at time t = b. YES
A) I only B) II only C) III only D) I and II only E) II and III only
    x  t    t  1  t  2      x '  t   3  t  1  t  2    t  1
                     3                                   2                     3



                                     x ' 1  3 1  1 1  2   1  1
                         3
    3 3  3        
                                                          2                        3
  x      1   2 
    2 2  2                     x '  2   3  2  1  2  2    2  1
                                                          2                            3
                                           f x
Let h  x   f g  x   and k  x  
                                                 . Fill in the chart below.
                                           g x
               x f(x) f ' (x) g(x) g ' (x) h(x) h ' (x)            k(x) k ' (x)
              -1 -1      4     1    -2      -1     8                -1
               0 1       0     0      0      1    0                 2      0
               1 1      -4     1     2      -1    -8                -1
h 1  f  g 1     h'  x   f ' g  x  g'  x 
                                             
  1  f 1          h'  1  f ' g  1 g'  1 h'  0   f ' g  0  g'  0 
                                                                            
h 0  f  g 0           8  f ' 1 g'  1          h'  0   f ' 0  g'  0 
h 0   f 0               8  4g'  1                h'  0    0  0 
                      h' 1  f ' g 1 g' 1
h 0  1                               
                         8  f ' 1 g' 1
                         8  4g' 1

                                                                  P. 225 #45 Finish

								
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