Derivative of $$$x^{4} - 6 x^{2}$$$

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

$${\color{red}\left(\frac{d}{dx} \left(x^{4} - 6 x^{2}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x^{4}\right) - \frac{d}{dx} \left(6 x^{2}\right)\right)}$$

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 $$$c = 6$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$$- {\color{red}\left(\frac{d}{dx} \left(6 x^{2}\right)\right)} + \frac{d}{dx} \left(x^{4}\right) = - {\color{red}\left(6 \frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(x^{4}\right)$$

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

$$- 6 {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(x^{4}\right) = - 6 {\color{red}\left(2 x\right)} + \frac{d}{dx} \left(x^{4}\right)$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 4$$$:

$$- 12 x + {\color{red}\left(\frac{d}{dx} \left(x^{4}\right)\right)} = - 12 x + {\color{red}\left(4 x^{3}\right)}$$

Simplify:

$$4 x^{3} - 12 x = 4 x \left(x^{2} - 3\right)$$

Thus, $$$\frac{d}{dx} \left(x^{4} - 6 x^{2}\right) = 4 x \left(x^{2} - 3\right)$$$.