Ableitung von $$$\operatorname{acos}{\left(1 - x^{4} \right)}$$$

Die Funktion $$$\operatorname{acos}{\left(1 - x^{4} \right)}$$$ ist die Komposition $$$f{\left(g{\left(x \right)} \right)}$$$ der beiden Funktionen $$$f{\left(u \right)} = \operatorname{acos}{\left(u \right)}$$$ und $$$g{\left(x \right)} = 1 - x^{4}$$$.

Wende die Kettenregel $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ an:

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

Die Ableitung des Arkuskosinus ist $$$\frac{d}{du} \left(\operatorname{acos}{\left(u \right)}\right) = - \frac{1}{\sqrt{1 - u^{2}}}$$$:

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

Zurück zur ursprünglichen Variable:

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

Die Ableitung einer Summe/Differenz ist die Summe/Differenz der Ableitungen:

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

Die Ableitung einer Konstante ist $$$0$$$:

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

Wende die Potenzregel $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ mit $$$n = 4$$$ an:

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

Vereinfachen:

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

Somit gilt $$$\frac{d}{dx} \left(\operatorname{acos}{\left(1 - x^{4} \right)}\right) = \frac{4 x}{\sqrt{2 - x^{4}}}$$$.