# Thread: Formal Definition of Derivative

1. ## Formal Definition of Derivative

I need to use the formal definition of a derivative,
g'(x) = lim h->0 (f(x + h) - f(x)) / h

to solve x^(2/3)...

lim h->0 (f(x+h)^(2/3) - x^(2/3)) / h

I know the answer is obviously (2/3)x^(-1/3) but we're supposed to use the difference quotient as stated above. Any help would be greatly appreciated

2. Originally Posted by DoQrs I need to use the formal definition of a derivative,
g'(x) = lim h->0 (f(x + h) - f(x)) / h

to solve x^(2/3)...

lim h->0 (f(x+h)^(2/3) - x^(2/3)) / h

I know the answer is obviously (2/3)x^(-1/3) but we're supposed to use the difference quotient as stated above. Any help would be greatly appreciated
$\displaystyle g(x) = x^{2/3}$

$\displaystyle \Rightarrow g'(x) = \lim_{h \to 0} \frac {g(x + h) - g(x)}h$

$\displaystyle = \lim_{h \to 0} \frac {(x + h)^{2/3} - x^{2/3}}h$

now simplify the top and find the limit. i believe rationalizing the top (multiplying by its conjugate over itself) should work

3. $\displaystyle \displaystyle g'(x)=\lim_{h\to 0}\frac{\sqrt{(x+h)^2}-\sqrt{x^2}}{h}=$
$\displaystyle \displaystyle=\lim_{h\to 0}\frac{(x+h)^2-x^2}{h\left(\sqrt{(x+h)^4}+\sqrt{x^2(x+h)^2}+\sqrt{x^4}\right)}=$
$\displaystyle \displaystyle=\lim_{h\to 0}\frac{h(2x+h)}{h\left(\sqrt{(x+h)^4}+\sqrt{x^2(x+h)^2}+\sqrt{x^4}\right)}=$
$\displaystyle \displaystyle=\frac{2x}{3\sqrt{x^4}}=\frac{2}{3\sqrt{x}}=\frac{2}{3}x^{-\frac{1}{3}}$

4. Thank you, did you use an online tool to generate the images?

5. We use Latex.
See "Latex Help" category, in the home page on this forum.

6. Originally Posted by DoQrs Thank you, did you use an online tool to generate the images?
or see our LaTex tutorial here

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