9 - Power Rule for Differentiation (6)

Suppose y=f(x)= x q  where q  is an irrational number. And, suppose { r k } k=1 k=  is an infinite sequence of rational numbers such that lim k r k =q . Then

dy dx = lim h0 f(x+h)f(x) h

= lim h0 (x+h) q x q h

= lim h0 (x+h) lim k r k x lim k r k h

= lim k lim h0 (x+h) r k x r k h

= lim k r k x r k 1

= lim k r k lim k x r k 1

=q x q1