P3.txt

[20pts] One definition of *derivative* is (inspired by [Rudin, 1964], p. 89):

    *Definition:* Let |f : [a, b] -> ℝ|.
    For an |x ∈ [a, b]|, consider the function |phi f (x) : [a, b] -> ℝ| by

<   phi f (x) (t)  =  (f(t) - f(x))/(t - x),    -- for |t /= x|

    and define

<   f'(x)  =  lim_{t->x} phi f (x)(t)

    provided that this limit exists.  We thus associate with |f| a
    function |f'| whose domain of definition is the set of points |x|
    at which the limit exists; |f'| is called the *derivative* of |f|.


    * [5pts] Let |r : [1, 2] -> ℝ| with |r(x) = 1/x|.
      Compute |r'| using this definition.

< phi r x t = (r(t)-r(x))/(t-x) = (1/t-1/x)/(t-x) = (x-t)/x/t/(t-x) = -1/x/t

Note that |-1/x/t| is defined also for |x=t| (as long as x/=0). Thus
the limit computation is trivial: |lim (phi r x) t = 1/x/t|

Thus we have:

  r' x = lim (phi r x) x = lim (\t->-1/x/t) x = -1/x^2

as expected.

    * [5pts] Let |h = g . f| for |f, g : [a, b] -> [a, b]|.
      Formulate the chain rule (the derivative of |h| in terms of
      operations on |f| and |g|).

I write |D f| for the derivative of |f| for clarity.

The chain rule: for |h = g . f| we have |D h = (D g . f)*D f|.

    * [10pts] Prove your formulation of the chain rule using the
      definition above.

  D h x
= -- def. of derivative
  lim (phi h x) x
= -- def. of h
  lim (phi (g . f) x) x

Let us do a sub-computation (prove a lemma) for arbitrary |t|:

  phi (g . f) x t
= -- def. of |phi|
  ((g . f)(t) - (g . f)(x))/(t - x)
= -- def. |.|
  (g(f t) - g(f x))/(t - x)
= -- Assume |f t - f x /= 0|
  (g(f t) - g(f x))/(f t - f x) * (f t - f x)/(t - x)
= -- def. of |phi|
  phi g (f x) (f t) * phi f x t

Now we can continue our equality proof:

  lim (phi (g . f) x) x
= -- Our new lemma
  lim (\t -> phi g (f x) (f t) * phi f x t) x
= -- limit of a product of functions is the product of the limits
  lim (\t -> phi g (f x) (f t)) x * lim (phi f x) x
= -- limit of a composition (using cont. of |f|) + def. of D f
  lim (\ft -> phi g (f x) ft) (f x) * D f x
= -- eta-reduction
  lim (phi g (f x)) (f x) * D f x
= -- def. of |D g| (at the point |f x|)
  D g (f x) * D f x
= -- Def. of |.| and |*| on functions
  ((D g . f) * D f) x

Thus we get |D h = (D g . f) * D f|.