Hi Omar,

You have $f(x) = x^{1/3}$ and you want to find its derivative directly from the definition of the derivative. You know that what you need is to find the difference quotient

\[\frac{f(x + h) - f(x)}{h}\]

and then take the limit as $h$ approaches zero.

First write

\[\frac{f(x + h) - f(x)}{h}\]

for $f(x) = x^{1/3}$ keeping in mind that $f(x+h) = (x+h)^{1/3}.$ Multiply the numerator and denominator of

\[\frac{f(x + h) - f(x)}{h}\]

by

\[\left(x+h\right)^{2/3} + x^{1/3}\left(x+h\right)^{1/3} + x^{2/3}.\]

Simplify.

The key to this problem is to write

\[h = (x+h) - x = \left( (x+h)^{1/3}\right)^3 - \left( x^{1/3}\right)^3\]

as a difference of cubes.

I hope this helps,
Penny