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Limit Definition of the Derivative

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					                          Harvey Mudd College Math Tutorial:

                Limit Definition of the Derivative
Once we know the most basic differentiation formulas and rules, we compute new derivatives
using what we already know. We rarely think back to where the basic formulas and rules
originated.

The geometric meaning of the derivative

                                                      df (x)
                                            f (x) =
                                                        dx
is the slope of the line tangent to y = f (x) at x.


Let’s look for this slope at P :

The secant line through P and Q has
slope

    f (x + ∆x) − f (x)   f (x + ∆x) − f (x)
                       =                    .
       (x + ∆x) − x             ∆x


We can approximate the tangent line through P by moving Q towards P , decreasing ∆x.
In the limit as ∆x → 0, we get the tangent line through P with slope

                                             f (x + ∆x) − f (x)
                                      lim                       .
                                     ∆x→0           ∆x
We define
                                            f (x + ∆x) − f (x) ∗
                                f (x) = lim                      .
                                       ∆x→0        ∆x

∗
    If the limit as ∆x → 0 at a particular point does not exist, f (x) is undefined at that point.

We derive all the basic differentiation formulas using this definition.


Example

For f (x) = x2 ,

                                           (x + ∆x)2 − x2
                           f (x) =     lim
                                      ∆x→0      ∆x
                                       (x2 + 2(∆x)x + ∆x2 ) − x2
                                =    lim
                                  ∆x→0           ∆x
                                       2(∆x)x + ∆x2
                                = lim
                                  ∆x→0      ∆x
                                = lim (2x + ∆x)
                                    ∆x→0
                                = 2x

as expected.


Example

              1
For f (x) =
              x
                                                     1     1
                                                   x+∆x
                                                       −   x
                                 f (x) =     lim
                                            ∆x→0      ∆x
                                                   x−(x+∆x)
                                                   (x+∆x)(x)
                                        =    lim
                                            ∆x→0     ∆x
                                                     −∆x
                                                   (x+∆x)(x)
                                        =    lim
                                            ∆x→0  ∆x
                                                   −1
                                        = lim
                                          ∆x→0 (x + ∆x)(x)
                                            1
                                        = − 2
                                            x
again as expected.


                                            Notes

The limit definition of the derivative is used to prove many well-known results, including the
following:


   • If f is differentiable at x0 , then f is continuous at x0 .
                                        d n
   • Differentiation of polynomials:       [x ] = nxn−1 .
                                       dx
   • Product and Quotient Rules for differentiation.
                                   Key Concepts

                       f (x + ∆x) − f (x)
We define f (x) = lim                      .
                ∆x→0          ∆x
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