Funktion $$$\sqrt{2} t - \sqrt{- \sqrt{2} \sqrt{\sqrt{5} + 3} - 2}$$$ derivaatta

Summan/erotuksen derivaatta on derivaattojen summa/erotus:

$${\color{red}\left(\frac{d}{dt} \left(\sqrt{2} t - \sqrt{- \sqrt{2} \sqrt{\sqrt{5} + 3} - 2}\right)\right)} = {\color{red}\left(\frac{d}{dt} \left(\sqrt{2} t\right) - \frac{d}{dt} \left(\sqrt{- \sqrt{2} \sqrt{\sqrt{5} + 3} - 2}\right)\right)}$$

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

$${\color{red}\left(\frac{d}{dt} \left(\sqrt{2} t\right)\right)} - \frac{d}{dt} \left(\sqrt{- \sqrt{2} \sqrt{\sqrt{5} + 3} - 2}\right) = {\color{red}\left(\sqrt{2} \frac{d}{dt} \left(t\right)\right)} - \frac{d}{dt} \left(\sqrt{- \sqrt{2} \sqrt{\sqrt{5} + 3} - 2}\right)$$

Vakion derivaatta on $$$0$$$:

$$- {\color{red}\left(\frac{d}{dt} \left(\sqrt{- \sqrt{2} \sqrt{\sqrt{5} + 3} - 2}\right)\right)} + \sqrt{2} \frac{d}{dt} \left(t\right) = - {\color{red}\left(0\right)} + \sqrt{2} \frac{d}{dt} \left(t\right)$$

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

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

Näin ollen, $$$\frac{d}{dt} \left(\sqrt{2} t - \sqrt{- \sqrt{2} \sqrt{\sqrt{5} + 3} - 2}\right) = \sqrt{2}$$$.