<s>[INST] Simplifying the derivative of $x^{\frac{2}{3}} \cdot (6-x)^{\frac{1}{3}}$ $$x^{\frac{2}{3}} \cdot (6-x)^{\frac{1}{3}}$$
So I get:
$$-x^{\frac{2}{3}} \cdot \frac{1}{3} (6-x) ^{\frac{-2}{3}} + (6-x) ^{\frac{1}{3}} \cdot \frac{2}{3} x ^ {\frac{-1}{3}}$$
How does one go about simplifying this?
I guess I can pull out common terms like this:
$$\frac{1}{3} x ^{-\frac{1}{3}} (6-x)^{\frac{-2}{3}} ( -x + (6-x) \cdot 2)$$
Is that right?
 [/INST] Alternatively:
$$\frac{d}{dx}\left(x^{\frac{2}{3}} \cdot (6-x)^{\frac{1}{3}}\right)=\frac{d}{dx}\left((6x^2-x^3)^{\frac13}\right)=\frac{1}{3}(6x^2-x^3)^{-\frac{2}{3}}(12x-3x^2)=\\
\frac{4x-x^2}{(6x^2-x^3)^{\frac{2}{3}}}=\frac{4x-x^2}{x^{\frac{4}{3}}(6-x)^{\frac23}}=\frac{4-x}{x^{\frac{1}{3}}(6-x)^{\frac23}}.$$
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