Revisiting Power Rule for any fractional exponents

Let y= x m n where n is a postive integer.

Note that any fraction a b can be written as m n where n is a postive integer.

Earlier we had fouind the derivative of y= x m n using the definition of the derivative and the fact that the power rule is true for both positive and integer exponents and for fractional exponents of the form 1 n where n is a postive integer. Here we will show the same except that instead of using the definition of the derivative, we will use the implicit differentiation.

y= x m n = ( x m ) 1 n

Differentiating with respect to x , we get

y= 1 n ( x m ) 1 n 1 m x m1 = m n x m n m x m1 = m n x m n m+m1 = m n x m n 1

:-)