$$$x$$$'e göre $$$\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}$$$'in türevi

Toplamın/farkın türevi, türevlerin toplamı/farkıdır:

$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)}\right) - \frac{d}{dx} \left(\sin{\left(x \right)} \cos{\left(a \right)}\right)\right)}$$

Sabit çarpan kuralını $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ $$$c = \cos{\left(a \right)}$$$ ve $$$f{\left(x \right)} = \sin{\left(x \right)}$$$ ile uygula:

$$- {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)} \cos{\left(a \right)}\right)\right)} + \frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)}\right) = - {\color{red}\left(\cos{\left(a \right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)}\right)$$

Sinüsün türevi $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:

$$- \cos{\left(a \right)} {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)}\right) = - \cos{\left(a \right)} {\color{red}\left(\cos{\left(x \right)}\right)} + \frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)}\right)$$

Sabit çarpan kuralını $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ $$$c = \sin{\left(a \right)}$$$ ve $$$f{\left(x \right)} = \cos{\left(x \right)}$$$ ile uygula:

$$- \cos{\left(a \right)} \cos{\left(x \right)} + {\color{red}\left(\frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)}\right)\right)} = - \cos{\left(a \right)} \cos{\left(x \right)} + {\color{red}\left(\sin{\left(a \right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$

Kosinüsün türevi $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:

$$\sin{\left(a \right)} {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} - \cos{\left(a \right)} \cos{\left(x \right)} = \sin{\left(a \right)} {\color{red}\left(- \sin{\left(x \right)}\right)} - \cos{\left(a \right)} \cos{\left(x \right)}$$

Sadeleştirin:

$$- \sin{\left(a \right)} \sin{\left(x \right)} - \cos{\left(a \right)} \cos{\left(x \right)} = - \cos{\left(a - x \right)}$$

Dolayısıyla, $$$\frac{d}{dx} \left(\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}\right) = - \cos{\left(a - x \right)}$$$.