1.7, The IRC of Power Functions

The main idea of this lecture is that we can compute the IRC of f(x) = x^n at an arbitrary point.

We do this by first assuming that the exponent n is a positive integer and writing out the formal definition of the derivative. According to that definition, we need to take the limit, as h goes to zero, of

f(x+h)-f(x)         (x+h)^n - x^n

-------------  =   -------------------

h                           h

We expand (x+h)^n by the binomial theorem (a.k.a Pascal's Triangle) and discover that x^n will cancel out, leaving us with an expression whose numerator begins

nx^(n-1)h +  (... higher powers of h...)

However, we must divide by h and then take the limit.  Dividing by h will give us

nx^(n-1) + h( ... more terms ...)

and when we evaluate the limit by setting h to zero, we get nx^(n-1).

It turns out that this formula works for any power of x, even if n is not an integer. (We won't prove that.)  Thus if

 f(x) = x^n

then the derivative is 

 f ' (x) = nx^(n-1).

This is our first derivative rule, often called the "power rule".

This lecture also gives us a change to teach (or review) Pascal's triangle.

The binomial theorem can be visualized by a triangle that begins with a ones at the top and has ones descending down the sides. We begin with row 0, which just has a 1 in it, followed by row 1 which is 1 1

As the triangle grows, the interior positions are filled by numbers created by adding the digits just above the number, on its left and right shoulders. Thus we row 2 with entries 1 2 1.

and after that, row 3 with entries  1 3 3 1

and so on.

 Here are rows 0 to 16!

(This version of Pascal's Triangle, created by Paul Gaborit (2009) under the Creative Commons attribution license, was found here.)