$$$\cos{\left(t \right)} - \cos{\left(2 t \right)}$$$の導関数

和/差の導関数は、導関数の和/差である:

$${\color{red}\left(\frac{d}{dt} \left(\cos{\left(t \right)} - \cos{\left(2 t \right)}\right)\right)} = {\color{red}\left(\frac{d}{dt} \left(\cos{\left(t \right)}\right) - \frac{d}{dt} \left(\cos{\left(2 t \right)}\right)\right)}$$

余弦関数の導関数は$$$\frac{d}{dt} \left(\cos{\left(t \right)}\right) = - \sin{\left(t \right)}$$$:

$${\color{red}\left(\frac{d}{dt} \left(\cos{\left(t \right)}\right)\right)} - \frac{d}{dt} \left(\cos{\left(2 t \right)}\right) = {\color{red}\left(- \sin{\left(t \right)}\right)} - \frac{d}{dt} \left(\cos{\left(2 t \right)}\right)$$

関数$$$\cos{\left(2 t \right)}$$$は、2つの関数$$$f{\left(u \right)} = \cos{\left(u \right)}$$$と$$$g{\left(t \right)} = 2 t$$$の合成$$$f{\left(g{\left(t \right)} \right)}$$$である。

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

$$- \sin{\left(t \right)} - {\color{red}\left(\frac{d}{dt} \left(\cos{\left(2 t \right)}\right)\right)} = - \sin{\left(t \right)} - {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{dt} \left(2 t\right)\right)}$$

余弦関数の導関数は$$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:

$$- \sin{\left(t \right)} - {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{dt} \left(2 t\right) = - \sin{\left(t \right)} - {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{dt} \left(2 t\right)$$

元の変数に戻す:

$$- \sin{\left(t \right)} + \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{dt} \left(2 t\right) = - \sin{\left(t \right)} + \sin{\left({\color{red}\left(2 t\right)} \right)} \frac{d}{dt} \left(2 t\right)$$

定数倍の法則 $$$\frac{d}{dt} \left(c f{\left(t \right)}\right) = c \frac{d}{dt} \left(f{\left(t \right)}\right)$$$ を $$$c = 2$$$ と $$$f{\left(t \right)} = t$$$ に対して適用します:

$$- \sin{\left(t \right)} + \sin{\left(2 t \right)} {\color{red}\left(\frac{d}{dt} \left(2 t\right)\right)} = - \sin{\left(t \right)} + \sin{\left(2 t \right)} {\color{red}\left(2 \frac{d}{dt} \left(t\right)\right)}$$

$$$n = 1$$$ を用いて冪法則 $$$\frac{d}{dt} \left(t^{n}\right) = n t^{n - 1}$$$ を適用すると、すなわち $$$\frac{d}{dt} \left(t\right) = 1$$$:

$$- \sin{\left(t \right)} + 2 \sin{\left(2 t \right)} {\color{red}\left(\frac{d}{dt} \left(t\right)\right)} = - \sin{\left(t \right)} + 2 \sin{\left(2 t \right)} {\color{red}\left(1\right)}$$

簡単化せよ:

$$- \sin{\left(t \right)} + 2 \sin{\left(2 t \right)} = \left(4 \cos{\left(t \right)} - 1\right) \sin{\left(t \right)}$$

したがって、$$$\frac{d}{dt} \left(\cos{\left(t \right)} - \cos{\left(2 t \right)}\right) = \left(4 \cos{\left(t \right)} - 1\right) \sin{\left(t \right)}$$$。