$$$\operatorname{acos}{\left(1 - x^{4} \right)}$$$ 的導數

函數 $$$\operatorname{acos}{\left(1 - x^{4} \right)}$$$ 是兩個函數 $$$f{\left(u \right)} = \operatorname{acos}{\left(u \right)}$$$ 與 $$$g{\left(x \right)} = 1 - x^{4}$$$ 之複合 $$$f{\left(g{\left(x \right)} \right)}$$$。

應用鏈式法則 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：

$${\color{red}\left(\frac{d}{dx} \left(\operatorname{acos}{\left(1 - x^{4} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\operatorname{acos}{\left(u \right)}\right) \frac{d}{dx} \left(1 - x^{4}\right)\right)}$$

反餘弦函數的導數為 $$$\frac{d}{du} \left(\operatorname{acos}{\left(u \right)}\right) = - \frac{1}{\sqrt{1 - u^{2}}}$$$：

$${\color{red}\left(\frac{d}{du} \left(\operatorname{acos}{\left(u \right)}\right)\right)} \frac{d}{dx} \left(1 - x^{4}\right) = {\color{red}\left(- \frac{1}{\sqrt{1 - u^{2}}}\right)} \frac{d}{dx} \left(1 - x^{4}\right)$$

返回原變數：

$$- \frac{\frac{d}{dx} \left(1 - x^{4}\right)}{\sqrt{1 - {\color{red}\left(u\right)}^{2}}} = - \frac{\frac{d}{dx} \left(1 - x^{4}\right)}{\sqrt{1 - {\color{red}\left(1 - x^{4}\right)}^{2}}}$$

和/差的導數等於導數的和/差：

$$- \frac{{\color{red}\left(\frac{d}{dx} \left(1 - x^{4}\right)\right)}}{\sqrt{1 - \left(1 - x^{4}\right)^{2}}} = - \frac{{\color{red}\left(\frac{d}{dx} \left(1\right) - \frac{d}{dx} \left(x^{4}\right)\right)}}{\sqrt{1 - \left(1 - x^{4}\right)^{2}}}$$

常數的導數為$$$0$$$：

$$- \frac{{\color{red}\left(\frac{d}{dx} \left(1\right)\right)} - \frac{d}{dx} \left(x^{4}\right)}{\sqrt{1 - \left(1 - x^{4}\right)^{2}}} = - \frac{{\color{red}\left(0\right)} - \frac{d}{dx} \left(x^{4}\right)}{\sqrt{1 - \left(1 - x^{4}\right)^{2}}}$$

套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 4$$$：

$$\frac{{\color{red}\left(\frac{d}{dx} \left(x^{4}\right)\right)}}{\sqrt{1 - \left(1 - x^{4}\right)^{2}}} = \frac{{\color{red}\left(4 x^{3}\right)}}{\sqrt{1 - \left(1 - x^{4}\right)^{2}}}$$

化簡：

$$\frac{4 x^{3}}{\sqrt{1 - \left(1 - x^{4}\right)^{2}}} = \frac{4 x}{\sqrt{2 - x^{4}}}$$

因此，$$$\frac{d}{dx} \left(\operatorname{acos}{\left(1 - x^{4} \right)}\right) = \frac{4 x}{\sqrt{2 - x^{4}}}$$$。