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									2.2 The derivative as a function
In Section 2.1 we considered the derivative at a fixed number a.
Now let number a vary. If we replace number a by a variable x,
then the derivative can be interpreted as a function of x :
                                  f ( x  h)  f ( x )
                 f ( x)  lim
                           h 0            h

Alternative notations for the derivative:
                        dy df   d
         f ( x)  y          f ( x)  Df ( x)
                        dx dx dx
D and d / dx are called differentiation operators.
dy / dx should not be regarded as a ratio.
                  4

                  3

                  2

                  1
                                      y  f  x

                  0    1     2    3      4   5     6   7   8   9
                  3
The derivative
is the slope of   2
the original
                  1
function.
                  0    1     2    3      4   5     6   7   8   9

                  -1
                           y  f  x
                  -2
         6
         5
         4
         3
                                     y  x 3  2



                                                                            
         2


                                          x  h
                                                    2
         1
                                                        3 x 3     2
-3 -2 -1 0
         -1
              1
                  x
                      2   3
                              y  lim
         -2                       h 0                   h
         -3
         6
         5
                                        x  2 xh  h  x
                                           2                 2           2
         4
         3
                              y  lim
                                   h 0         h
         2
         1
                                                                     0
                                         y  lim 2 x  h
-3 -2 -1 0    1 2 3
        -1     x
        -2                                     h0
        -3

                                               y  2 x
        -4
        -5
        -6
            Differentiable functions

•A function f is differentiable at a if f ′(a) exists.
It is differentiable on an open interval (a,b) [ or
(a,) or (- , a) or (- , ) ] if it is differentiable
at every number in the interval.

•Theorem: If f is differentiable at a, then f is
continuous at a.

• Note: The converse is false: there are functions
that are continuous but not differentiable.
      Example: f(x) = | x |
   To be differentiable, a function must be continuous and
   smooth.

    Derivatives will fail to exist at:



f  x  x
                                                                         2
                                                            f  x  x   3




                   corner                    cusp




                                                             1, x  0
                                                    f  x  
f  x  3 x                                                  1, x  0
               vertical tangent          discontinuity
                                                                     
                Higher Order Derivatives:
     dy
y              is the first derivative of y with respect to x.
     dx

      dy d dy d 2 y                is the second derivative.
y          2
      dx dx dx dx                     (y double prime)

       dy
y                 is the third derivative.    We will learn
       dx                                         later what these
                                                  higher order
             d                                    derivatives are
     4
y              y is the fourth derivative.    used for.
             dx

								
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