Show how to find the derivative of the cube root of x without using the power rule for differentiation.  (i.e. use the four-step process to find f’(x) if f(x) = x1/3.)

Expert Answers
hala718 eNotes educator| Certified Educator

`f(x)= x^(1/3)`

We can use the implicit differentiation to find the derivative

==>  `y= x^(1/3)`

Cube both sides

==> `y^3 = x`

Now use implicit differentiation.

==> `3y^2 y' = 1`

Now we will divide by `3y^2` .

==> `y'= 1/(3y^2)`

Now we will substitute with `y= x^(1/3)`

`==gt y'= 1/(3(x^(1/3))^2) `

`==gt y'= 1/(3x^(2/3))`