On-line Math 21

2.2  Differentiation Rules

Theorem 2 [The Power Rule]. For each positive integer n , (xⁿ)¢ = nx^n-1.

Proof:

This works similarly to the specific cases we went over earlier. Audio

For f(x) = xⁿ ,

f¢(x) = lim h® 0  f(x+h)-f(x) h = lim h® 0  (x+h)n-(x)n h = lim h® 0  xn+n xn-1h+... +hn-xn h  Hint = lim h® 0  n xn-1+... +hn-1, by cancelling the xn¢s and dividing by h. = n xn-1.

Copyright (c) 2000 by David L. Johnson.