How do I do this?
(1/ 3x^(2/3)) + (2/3x^(5/3))
thanks
I guess this is the exercice:
$\displaystyle \frac{1}{3\cdot x^{\frac{2}{3}}}+\frac{2}{3\cdot x^{\frac{5}{3}}}$
?
Write:
$\displaystyle 3x^{\frac{5}{3}}=3x^{\frac{2}{3}}.x^{\frac{3}{3}}= 3x^{\frac{2}{3}}.x$
Can you now finish it? ...
Yes, but I edited my post, because there was a mistake in it.
I just used the basic rule:
$\displaystyle a^x\cdot a^y=a^{x+y}$
It also important, is it:
$\displaystyle (3x)^{\frac{5}{3}}$ or $\displaystyle 3\cdot x^{\frac{5}{3}}$
I think the second one. Can you confirm this? ...
You made a mistake in your derivative, but in your first post it was correct so I guess you didn't see it:
$\displaystyle D\left(x^{\frac{1}{3}}\right)=\frac{1}{3}x^{\frac{-2}{3}}$
So you get indeed:
$\displaystyle \frac{1}{3\cdot x^{\frac{2}{3}}}+\frac{2}{3\cdot x^{\frac{5}{3}}}=\frac{1}{3} \left(\frac{1}{x^{\frac{2}{3}}}+\frac{2}{x^{\frac{ 5}{3}}} \right)$
Now, you can wright, like I said before:
$\displaystyle x^{\frac{5}{3}}=x^{\frac{2}{3}}\cdot x$
So you get:
$\displaystyle \frac{1}{3} \left(\frac{1}{x^{\frac{2}{3}}}+\frac{2}{x^{\frac{ 2}{3}}\cdot x}\right)$
Now if you multiply the left fraction with $\displaystyle x$ in the numerator and denominator you'll get the same denominator as the right fraction, so:
$\displaystyle ...=\frac{1}{3} \left(\frac{x}{x^{\frac{5}{3}}}+\frac{2}{x^{\frac{ 5}{3}}}\right)=\frac{2+x}{3\cdot x^{\frac{5}{3}}}$