$$$\operatorname{acos}{\left(1 - x^{4} \right)}$$$の導関数

関数$$$\operatorname{acos}{\left(1 - x^{4} \right)}$$$は、2つの関数$$$f{\left(u \right)} = \operatorname{acos}{\left(u \right)}$$$と$$$g{\left(x \right)} = 1 - x^{4}$$$の合成$$$f{\left(g{\left(x \right)} \right)}$$$である。

連鎖律 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ を適用する:

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

逆余弦関数の導関数は$$$\frac{d}{du} \left(\operatorname{acos}{\left(u \right)}\right) = - \frac{1}{\sqrt{1 - u^{2}}}$$$です:

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

元の変数に戻す:

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

和/差の導関数は、導関数の和/差である:

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

定数の導数は$$$0$$$です:

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

冪法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ を $$$n = 4$$$ に対して適用する:

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

簡単化せよ:

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

したがって、$$$\frac{d}{dx} \left(\operatorname{acos}{\left(1 - x^{4} \right)}\right) = \frac{4 x}{\sqrt{2 - x^{4}}}$$$。