Funktion $$$\frac{1}{\sqrt{5 t^{2} + 1}}$$$ derivaatta

Funktio $$$\frac{1}{\sqrt{5 t^{2} + 1}}$$$ on kahden funktion $$$f{\left(u \right)} = \frac{1}{\sqrt{u}}$$$ ja $$$g{\left(t \right)} = 5 t^{2} + 1$$$ yhdistelmä $$$f{\left(g{\left(t \right)} \right)}$$$.

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

$${\color{red}\left(\frac{d}{dt} \left(\frac{1}{\sqrt{5 t^{2} + 1}}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\frac{1}{\sqrt{u}}\right) \frac{d}{dt} \left(5 t^{2} + 1\right)\right)}$$

Sovella potenssisääntöä $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$, kun $$$n = - \frac{1}{2}$$$:

$${\color{red}\left(\frac{d}{du} \left(\frac{1}{\sqrt{u}}\right)\right)} \frac{d}{dt} \left(5 t^{2} + 1\right) = {\color{red}\left(- \frac{1}{2 u^{\frac{3}{2}}}\right)} \frac{d}{dt} \left(5 t^{2} + 1\right)$$

Palaa alkuperäiseen muuttujaan:

$$- \frac{\frac{d}{dt} \left(5 t^{2} + 1\right)}{2 {\color{red}\left(u\right)}^{\frac{3}{2}}} = - \frac{\frac{d}{dt} \left(5 t^{2} + 1\right)}{2 {\color{red}\left(5 t^{2} + 1\right)}^{\frac{3}{2}}}$$

Summan/erotuksen derivaatta on derivaattojen summa/erotus:

$$- \frac{{\color{red}\left(\frac{d}{dt} \left(5 t^{2} + 1\right)\right)}}{2 \left(5 t^{2} + 1\right)^{\frac{3}{2}}} = - \frac{{\color{red}\left(\frac{d}{dt} \left(5 t^{2}\right) + \frac{d}{dt} \left(1\right)\right)}}{2 \left(5 t^{2} + 1\right)^{\frac{3}{2}}}$$

Vakion derivaatta on $$$0$$$:

$$- \frac{{\color{red}\left(\frac{d}{dt} \left(1\right)\right)} + \frac{d}{dt} \left(5 t^{2}\right)}{2 \left(5 t^{2} + 1\right)^{\frac{3}{2}}} = - \frac{{\color{red}\left(0\right)} + \frac{d}{dt} \left(5 t^{2}\right)}{2 \left(5 t^{2} + 1\right)^{\frac{3}{2}}}$$

Sovella vakion kerroinsääntöä $$$\frac{d}{dt} \left(c f{\left(t \right)}\right) = c \frac{d}{dt} \left(f{\left(t \right)}\right)$$$ käyttäen $$$c = 5$$$ ja $$$f{\left(t \right)} = t^{2}$$$:

$$- \frac{{\color{red}\left(\frac{d}{dt} \left(5 t^{2}\right)\right)}}{2 \left(5 t^{2} + 1\right)^{\frac{3}{2}}} = - \frac{{\color{red}\left(5 \frac{d}{dt} \left(t^{2}\right)\right)}}{2 \left(5 t^{2} + 1\right)^{\frac{3}{2}}}$$

Sovella potenssisääntöä $$$\frac{d}{dt} \left(t^{n}\right) = n t^{n - 1}$$$, kun $$$n = 2$$$:

$$- \frac{5 {\color{red}\left(\frac{d}{dt} \left(t^{2}\right)\right)}}{2 \left(5 t^{2} + 1\right)^{\frac{3}{2}}} = - \frac{5 {\color{red}\left(2 t\right)}}{2 \left(5 t^{2} + 1\right)^{\frac{3}{2}}}$$

Näin ollen, $$$\frac{d}{dt} \left(\frac{1}{\sqrt{5 t^{2} + 1}}\right) = - \frac{5 t}{\left(5 t^{2} + 1\right)^{\frac{3}{2}}}$$$.