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 Question from omar, a student: hi can help me am teacher ask me about x^1/3 Derivation definition .

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

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