4 - Power Rule for Differentiation (3)

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

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 n x 1 n ][ (x+h ) n1 n + (x+h ) n2 n x 1 n + (x+h ) n3 n x 2 n ++ (x+h ) 1 n x n2 n + x n1 n ] h [ (x+h ) n1 n + (x+h ) n2 n x 1 n + (x+h ) n3 n x 2 n + + (x+h ) 1 n x n2 n + x n1 n ]

= lim h0 (x+h )x h [ (x+h ) n1 n + (x+h ) n2 n x 1 n + (x+h ) n3 n x 2 n + + (x+h ) 1 n x n2 n + x n1 n ]

= lim h0 1 (x+h ) n1 n + (x+h ) n2 n x 1 n + (x+h ) n3 n x 2 n ++ (x+h ) 1 n x n2 n + x n1 n

= 1 n x n1 n

= 1 n x 1n n = 1 n x 1n n

= 1 n x 1 n 1