5 - Power Rule for Differentiation (4)

Let y=f(x)= x 1 n  , where n  is a negative integer. Then n=m  , where m  is a positive integer. So,

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

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

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

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

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

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

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

= 1 m x 1 m 1 1 x 1 m x 1 m

= 1 m x 1 m 1

= 1 n x 1 n 1