3 - Power Rule for Differentiation (2)

Let y=f(x)= x n  , where n  is a negative integer. Then x n = x m  , where m  is a positive integer. So,

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

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

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

= lim h0 1 (x+h) m 1 x m h

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

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

= lim h0 (x+h) m x m h 1 (x+h) m ( x m )

=m x m1 1 x m x m

=m x m1

=n x n1