Funktion $$$2 t - 1 + \frac{1}{t}$$$ derivaatta

Summan/erotuksen derivaatta on derivaattojen summa/erotus:

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

Vakion derivaatta on $$$0$$$:

$$- {\color{red}\left(\frac{d}{dt} \left(1\right)\right)} + \frac{d}{dt} \left(\frac{1}{t}\right) + \frac{d}{dt} \left(2 t\right) = - {\color{red}\left(0\right)} + \frac{d}{dt} \left(\frac{1}{t}\right) + \frac{d}{dt} \left(2 t\right)$$

Sovella vakion kerroinsääntöä $$$\frac{d}{dt} \left(c f{\left(t \right)}\right) = c \frac{d}{dt} \left(f{\left(t \right)}\right)$$$ käyttäen $$$c = 2$$$ ja $$$f{\left(t \right)} = t$$$:

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

Sovella potenssisääntöä $$$\frac{d}{dt} \left(t^{n}\right) = n t^{n - 1}$$$ käyttäen $$$n = 1$$$, toisin sanoen, $$$\frac{d}{dt} \left(t\right) = 1$$$:

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

Sovella potenssisääntöä $$$\frac{d}{dt} \left(t^{n}\right) = n t^{n - 1}$$$, kun $$$n = -1$$$:

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

Näin ollen, $$$\frac{d}{dt} \left(2 t - 1 + \frac{1}{t}\right) = 2 - \frac{1}{t^{2}}$$$.