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Παράγωγος της $$$\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}$$$ ως προς $$$x$$$
Σχετικοί υπολογιστές: Υπολογιστής λογαριθμικής παραγώγισης, Υπολογιστής Έμμεσης Παραγώγισης με Βήματα
Η είσοδός σας
Βρείτε $$$\frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}\right)$$$.
Λύση
Η παράγωγος του αθροίσματος/της διαφοράς είναι το άθροισμα/η διαφορά των παραγώγων:
$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)}\right) - \frac{d}{dx} \left(\sin{\left(x \right)} \cos{\left(a \right)}\right)\right)}$$Εφαρμόστε τον κανόνα του σταθερού πολλαπλασιαστή $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ με $$$c = \cos{\left(a \right)}$$$ και $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)} \cos{\left(a \right)}\right)\right)} + \frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)}\right) = - {\color{red}\left(\cos{\left(a \right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)}\right)$$Η παράγωγος του ημιτόνου είναι $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$- \cos{\left(a \right)} {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)}\right) = - \cos{\left(a \right)} {\color{red}\left(\cos{\left(x \right)}\right)} + \frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)}\right)$$Εφαρμόστε τον κανόνα του σταθερού πολλαπλασιαστή $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ με $$$c = \sin{\left(a \right)}$$$ και $$$f{\left(x \right)} = \cos{\left(x \right)}$$$:
$$- \cos{\left(a \right)} \cos{\left(x \right)} + {\color{red}\left(\frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)}\right)\right)} = - \cos{\left(a \right)} \cos{\left(x \right)} + {\color{red}\left(\sin{\left(a \right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$Η παράγωγος του συνημιτόνου είναι $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:
$$\sin{\left(a \right)} {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} - \cos{\left(a \right)} \cos{\left(x \right)} = \sin{\left(a \right)} {\color{red}\left(- \sin{\left(x \right)}\right)} - \cos{\left(a \right)} \cos{\left(x \right)}$$Απλοποιήστε:
$$- \sin{\left(a \right)} \sin{\left(x \right)} - \cos{\left(a \right)} \cos{\left(x \right)} = - \cos{\left(a - x \right)}$$Άρα, $$$\frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}\right) = - \cos{\left(a - x \right)}$$$.
Απάντηση
$$$\frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}\right) = - \cos{\left(a - x \right)}$$$A