6 - Power Rule for Differentiation (5)

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

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

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

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

= lim h0 [ (x+h) 1 n x 1 n ][ ( (x+h) 1 n ) m1 + ( (x+h) 1 n ) m2 x 1 n ++ (x+h) 1 n ( x 1 n ) m2 + ( x 1 n ) m1 ] h

= lim h0 (x+h ) 1 n x 1 n h [ ( (x+h) 1 n ) m1 + ( (x+h) 1 n ) m2 x 1 n ++ (x+h) 1 n ( x 1 n ) m2 + ( x 1 n ) m1 ]

= 1 n x 1 n 1 [m x m1 n ]

= m n x m n 1