Derivatan av $$$\cos{\left(t \right)} - \cos{\left(2 t \right)}$$$

Derivatan av en summa/differens är summan/differensen av derivatorna:

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

Funktionen $$$\cos{\left(2 t \right)}$$$ är sammansättningen $$$f{\left(g{\left(t \right)} \right)}$$$ av två funktioner $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ och $$$g{\left(t \right)} = 2 t$$$.

Tillämpa kedjeregeln $$$\frac{d}{dt} \left(f{\left(g{\left(t \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dt} \left(g{\left(t \right)}\right)$$$:

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

Derivatan av cosinus är $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:

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

Återgå till den ursprungliga variabeln:

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

Tillämpa konstantfaktorregeln $$$\frac{d}{dt} \left(c f{\left(t \right)}\right) = c \frac{d}{dt} \left(f{\left(t \right)}\right)$$$ med $$$c = 2$$$ och $$$f{\left(t \right)} = t$$$:

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

Tillämpa potensregeln $$$\frac{d}{dt} \left(t^{n}\right) = n t^{n - 1}$$$ med $$$n = 1$$$, det vill säga $$$\frac{d}{dt} \left(t\right) = 1$$$:

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

Derivatan av cosinus är $$$\frac{d}{dt} \left(\cos{\left(t \right)}\right) = - \sin{\left(t \right)}$$$:

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

Förenkla:

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

Alltså, $$$\frac{d}{dt} \left(\cos{\left(t \right)} - \cos{\left(2 t \right)}\right) = \left(4 \cos{\left(t \right)} - 1\right) \sin{\left(t \right)}$$$.