The Derivative

							The derivative as the slope of the
          tangent line

             (at a point)
      What is a derivative?

• A function
• the rate of change of a function
• the slope of the line tangent to
  the curve
The tangent line




                   single point
                   of intersection
     slope of a secant line

                            f(a) - f(x)
                              a-x




    f(x)


                     f(a)
x                              a
slope of a (closer) secant line

                            f(a) - f(x)
                              a-x




      f(x)

                     f(a)
  x          x                 a
closer and closer…




                 a
watch the slope...
    watch what x does...




x                     a
The slope of the secant line gets closer and closer to
           the slope of the tangent line...
As the values of x get closer and closer to a!




   x                                 a
  The slope of the secant lines
            gets closer
to the slope of the tangent line...

        ...as the values of x
            get closer to a

          Translates to….
            lim          f(x) - f(a)
            x    a          x-a

as x goes to a
                                Equation for the slope


Which gives us the the exact slope
of the line tangent to the curve at a!
               similarly...

                                             f(x+h) - f(x)
                                               (x+h) - x

                                                           = f(x+h) - f(x)
                                                                  h

 f(a+h)
                    h

                                           f(a)
a+h                                                    a
  (For this particular curve, h is a negative value)
                         thus...

    lim f(a+h) - f(a)
    h 0
             h

                       AND


                             lim f(x) - f(a)
                             x   a     x-a

Give us a way to calculate the slope of the line tangent at a!
Which one should I use?

     (doesn’t really matter)
 A VERY simple example...
yx   2




            y  x2




                     want the slope
                     where a=2
    f (x)  f (a)       x a
                          2     2
                                   (x  a)(x  a)
lim                lim       lim
        xa              xa            xa

        lim( x  a)  lim( x  2)  4



                   as x       a=2
    f ( x  h)  f ( x )       ( x  h) 2  x 2
lim                       lim
             h                        h

      x  2 xh  h  x
        2             2     2
                             h( 2 x  h)
 lim                   lim
              h                   h

             lim( 2 x  h)  4


              As h    0
          back to our example...
yx   2




                    y  x2




                             When a=2,
                             the slope is 4
            In conclusion...
• The derivative is the the slope of the line
  tangent to the curve (evaluated at a point).
  Now you know why slope was so important
  in your algebra classes!
• It is a limit (2 ways to define it).
• Once you learn the rules of derivatives, you
  will still remember these limit definitions,
  but you won’t use them to figure out limits.
Derivative of a constant function:

The derivative of f(x) = c where c is a constant is
given by 
f '(x) = 0



Example 

f(x) = - 10

f '(x) = 0 

Derivative of a power function (power rule):

The derivative of f(x) = x r where r is a constant real
number is given by 
f '(x) = r x r - 1


Example 

f(x) = x -2

f '(x) = -2 x -3 = -2 / x 3
Derivative of a function multiplied by a constant:

The derivative of f(x) = c g(x) is given by 
f '(x) =
c g '(x)


Example

f(x) = 3x 3 , c = 3, and g(x) = x 3

f '(x) = c g '(x) = 3 (3x 2) = 9 x 2
Derivative of the sum of functions (sum rule):

The derivative of f(x) = g(x) + h(x) is given by 
f
'(x) = g '(x) + h '(x)


Example 
f(x) = x 2 + 4 

Let g(x) = x 2 and h(x)
= 4,
then f '(x) = g '(x) + h '(x) = 2x + 0 = 2x