Derivatan av $$$\frac{7}{\sqrt{9 t^{4} + 4 t^{2} + 49}}$$$

Tillämpa konstantfaktorregeln $$$\frac{d}{dt} \left(c f{\left(t \right)}\right) = c \frac{d}{dt} \left(f{\left(t \right)}\right)$$$ med $$$c = 7$$$ och $$$f{\left(t \right)} = \frac{1}{\sqrt{9 t^{4} + 4 t^{2} + 49}}$$$:

$${\color{red}\left(\frac{d}{dt} \left(\frac{7}{\sqrt{9 t^{4} + 4 t^{2} + 49}}\right)\right)} = {\color{red}\left(7 \frac{d}{dt} \left(\frac{1}{\sqrt{9 t^{4} + 4 t^{2} + 49}}\right)\right)}$$

Funktionen $$$\frac{1}{\sqrt{9 t^{4} + 4 t^{2} + 49}}$$$ är sammansättningen $$$f{\left(g{\left(t \right)} \right)}$$$ av två funktioner $$$f{\left(u \right)} = \frac{1}{\sqrt{u}}$$$ och $$$g{\left(t \right)} = 9 t^{4} + 4 t^{2} + 49$$$.

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

$$7 {\color{red}\left(\frac{d}{dt} \left(\frac{1}{\sqrt{9 t^{4} + 4 t^{2} + 49}}\right)\right)} = 7 {\color{red}\left(\frac{d}{du} \left(\frac{1}{\sqrt{u}}\right) \frac{d}{dt} \left(9 t^{4} + 4 t^{2} + 49\right)\right)}$$

Tillämpa potensregeln $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ med $$$n = - \frac{1}{2}$$$:

$$7 {\color{red}\left(\frac{d}{du} \left(\frac{1}{\sqrt{u}}\right)\right)} \frac{d}{dt} \left(9 t^{4} + 4 t^{2} + 49\right) = 7 {\color{red}\left(- \frac{1}{2 u^{\frac{3}{2}}}\right)} \frac{d}{dt} \left(9 t^{4} + 4 t^{2} + 49\right)$$

Återgå till den ursprungliga variabeln:

$$- \frac{7 \frac{d}{dt} \left(9 t^{4} + 4 t^{2} + 49\right)}{2 {\color{red}\left(u\right)}^{\frac{3}{2}}} = - \frac{7 \frac{d}{dt} \left(9 t^{4} + 4 t^{2} + 49\right)}{2 {\color{red}\left(9 t^{4} + 4 t^{2} + 49\right)}^{\frac{3}{2}}}$$

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

$$- \frac{7 {\color{red}\left(\frac{d}{dt} \left(9 t^{4} + 4 t^{2} + 49\right)\right)}}{2 \left(9 t^{4} + 4 t^{2} + 49\right)^{\frac{3}{2}}} = - \frac{7 {\color{red}\left(\frac{d}{dt} \left(9 t^{4}\right) + \frac{d}{dt} \left(4 t^{2}\right) + \frac{d}{dt} \left(49\right)\right)}}{2 \left(9 t^{4} + 4 t^{2} + 49\right)^{\frac{3}{2}}}$$

Tillämpa konstantfaktorregeln $$$\frac{d}{dt} \left(c f{\left(t \right)}\right) = c \frac{d}{dt} \left(f{\left(t \right)}\right)$$$ med $$$c = 4$$$ och $$$f{\left(t \right)} = t^{2}$$$:

$$- \frac{7 \left({\color{red}\left(\frac{d}{dt} \left(4 t^{2}\right)\right)} + \frac{d}{dt} \left(49\right) + \frac{d}{dt} \left(9 t^{4}\right)\right)}{2 \left(9 t^{4} + 4 t^{2} + 49\right)^{\frac{3}{2}}} = - \frac{7 \left({\color{red}\left(4 \frac{d}{dt} \left(t^{2}\right)\right)} + \frac{d}{dt} \left(49\right) + \frac{d}{dt} \left(9 t^{4}\right)\right)}{2 \left(9 t^{4} + 4 t^{2} + 49\right)^{\frac{3}{2}}}$$

Tillämpa potensregeln $$$\frac{d}{dt} \left(t^{n}\right) = n t^{n - 1}$$$ med $$$n = 2$$$:

$$- \frac{7 \left(4 {\color{red}\left(\frac{d}{dt} \left(t^{2}\right)\right)} + \frac{d}{dt} \left(49\right) + \frac{d}{dt} \left(9 t^{4}\right)\right)}{2 \left(9 t^{4} + 4 t^{2} + 49\right)^{\frac{3}{2}}} = - \frac{7 \left(4 {\color{red}\left(2 t\right)} + \frac{d}{dt} \left(49\right) + \frac{d}{dt} \left(9 t^{4}\right)\right)}{2 \left(9 t^{4} + 4 t^{2} + 49\right)^{\frac{3}{2}}}$$

Derivatan av en konstant är $$$0$$$:

$$- \frac{7 \left(8 t + {\color{red}\left(\frac{d}{dt} \left(49\right)\right)} + \frac{d}{dt} \left(9 t^{4}\right)\right)}{2 \left(9 t^{4} + 4 t^{2} + 49\right)^{\frac{3}{2}}} = - \frac{7 \left(8 t + {\color{red}\left(0\right)} + \frac{d}{dt} \left(9 t^{4}\right)\right)}{2 \left(9 t^{4} + 4 t^{2} + 49\right)^{\frac{3}{2}}}$$

Tillämpa konstantfaktorregeln $$$\frac{d}{dt} \left(c f{\left(t \right)}\right) = c \frac{d}{dt} \left(f{\left(t \right)}\right)$$$ med $$$c = 9$$$ och $$$f{\left(t \right)} = t^{4}$$$:

$$- \frac{7 \left(8 t + {\color{red}\left(\frac{d}{dt} \left(9 t^{4}\right)\right)}\right)}{2 \left(9 t^{4} + 4 t^{2} + 49\right)^{\frac{3}{2}}} = - \frac{7 \left(8 t + {\color{red}\left(9 \frac{d}{dt} \left(t^{4}\right)\right)}\right)}{2 \left(9 t^{4} + 4 t^{2} + 49\right)^{\frac{3}{2}}}$$

Tillämpa potensregeln $$$\frac{d}{dt} \left(t^{n}\right) = n t^{n - 1}$$$ med $$$n = 4$$$:

$$- \frac{7 \left(8 t + 9 {\color{red}\left(\frac{d}{dt} \left(t^{4}\right)\right)}\right)}{2 \left(9 t^{4} + 4 t^{2} + 49\right)^{\frac{3}{2}}} = - \frac{7 \left(8 t + 9 {\color{red}\left(4 t^{3}\right)}\right)}{2 \left(9 t^{4} + 4 t^{2} + 49\right)^{\frac{3}{2}}}$$

Förenkla:

$$- \frac{7 \left(36 t^{3} + 8 t\right)}{2 \left(9 t^{4} + 4 t^{2} + 49\right)^{\frac{3}{2}}} = - \frac{14 t \left(9 t^{2} + 2\right)}{\left(9 t^{4} + 4 t^{2} + 49\right)^{\frac{3}{2}}}$$

Alltså, $$$\frac{d}{dt} \left(\frac{7}{\sqrt{9 t^{4} + 4 t^{2} + 49}}\right) = - \frac{14 t \left(9 t^{2} + 2\right)}{\left(9 t^{4} + 4 t^{2} + 49\right)^{\frac{3}{2}}}$$$.