



 
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,  


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