Ableitung von $$$\sin{\left(x \right)} - \cos{\left(x \right)}$$$
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Ihre Eingabe
Bestimme $$$\frac{d}{dx} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)$$$.
Lösung
Die Ableitung einer Summe/Differenz ist die Summe/Differenz der Ableitungen:
$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right) - \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$Die Ableitung des Kosinus ist $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(\sin{\left(x \right)}\right) = - {\color{red}\left(- \sin{\left(x \right)}\right)} + \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$Die Ableitung des Sinus ist $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$\sin{\left(x \right)} + {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = \sin{\left(x \right)} + {\color{red}\left(\cos{\left(x \right)}\right)}$$Vereinfachen:
$$\sin{\left(x \right)} + \cos{\left(x \right)} = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}$$Somit gilt $$$\frac{d}{dx} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}$$$.
Antwort
$$$\frac{d}{dx} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}$$$A