Derivative of $$$\sqrt[9]{x^{2} + 3 + \frac{2}{x^{2}}}$$$

The function $$$\sqrt[9]{x^{2} + 3 + \frac{2}{x^{2}}}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \sqrt[9]{u}$$$ and $$$g{\left(x \right)} = x^{2} + 3 + \frac{2}{x^{2}}$$$.

Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(\sqrt[9]{x^{2} + 3 + \frac{2}{x^{2}}}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sqrt[9]{u}\right) \frac{d}{dx} \left(x^{2} + 3 + \frac{2}{x^{2}}\right)\right)}$$

Apply the power rule $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ with $$$n = \frac{1}{9}$$$:

$${\color{red}\left(\frac{d}{du} \left(\sqrt[9]{u}\right)\right)} \frac{d}{dx} \left(x^{2} + 3 + \frac{2}{x^{2}}\right) = {\color{red}\left(\frac{1}{9 u^{\frac{8}{9}}}\right)} \frac{d}{dx} \left(x^{2} + 3 + \frac{2}{x^{2}}\right)$$

Return to the old variable:

$$\frac{\frac{d}{dx} \left(x^{2} + 3 + \frac{2}{x^{2}}\right)}{9 {\color{red}\left(u\right)}^{\frac{8}{9}}} = \frac{\frac{d}{dx} \left(x^{2} + 3 + \frac{2}{x^{2}}\right)}{9 {\color{red}\left(x^{2} + 3 + \frac{2}{x^{2}}\right)}^{\frac{8}{9}}}$$

The derivative of a sum/difference is the sum/difference of derivatives:

$$\frac{{\color{red}\left(\frac{d}{dx} \left(x^{2} + 3 + \frac{2}{x^{2}}\right)\right)}}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}} = \frac{{\color{red}\left(\frac{d}{dx} \left(x^{2}\right) + \frac{d}{dx} \left(3\right) + \frac{d}{dx} \left(\frac{2}{x^{2}}\right)\right)}}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}}$$

The derivative of a constant is $$$0$$$:

$$\frac{{\color{red}\left(\frac{d}{dx} \left(3\right)\right)} + \frac{d}{dx} \left(\frac{2}{x^{2}}\right) + \frac{d}{dx} \left(x^{2}\right)}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}} = \frac{{\color{red}\left(0\right)} + \frac{d}{dx} \left(\frac{2}{x^{2}}\right) + \frac{d}{dx} \left(x^{2}\right)}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 2$$$:

$$\frac{{\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(\frac{2}{x^{2}}\right)}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}} = \frac{{\color{red}\left(2 x\right)} + \frac{d}{dx} \left(\frac{2}{x^{2}}\right)}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}}$$

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 2$$$ and $$$f{\left(x \right)} = \frac{1}{x^{2}}$$$:

$$\frac{2 x + {\color{red}\left(\frac{d}{dx} \left(\frac{2}{x^{2}}\right)\right)}}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}} = \frac{2 x + {\color{red}\left(2 \frac{d}{dx} \left(\frac{1}{x^{2}}\right)\right)}}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = -2$$$:

$$\frac{2 x + 2 {\color{red}\left(\frac{d}{dx} \left(\frac{1}{x^{2}}\right)\right)}}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}} = \frac{2 x + 2 {\color{red}\left(- \frac{2}{x^{3}}\right)}}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}}$$

Thus, $$$\frac{d}{dx} \left(\sqrt[9]{x^{2} + 3 + \frac{2}{x^{2}}}\right) = \frac{2 x - \frac{4}{x^{3}}}{9 \left(x^{2} + 3 + \frac{2}{x^{2}}\right)^{\frac{8}{9}}}$$$.