The derivative of a Qoutient

$\mathit{F}\mathit{i}\mathit{n}\mathit{d}\mathbf{ }\mathit{t}\mathit{h}\mathit{e}\mathbf{ }\mathit{D}\mathit{e}\mathit{r}\mathit{i}\mathit{v}\mathit{a}\mathit{t}\mathit{i}\mathit{v}\mathit{e}\mathbf{ }\mathit{o}\mathit{f}\mathbf{:}\mathbf{ }\mathit{y}\mathbf{=}\frac{\mathbf{3}}{{\left({\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\right)}^{\mathbf{3}}}$

$\mathit{S}\mathit{t}\mathit{e}\mathit{p}\mathit{ }\mathit{1}\mathit{:}\mathit{ }\mathit{M}\mathit{u}\mathit{l}\mathit{t}\mathit{i}\mathit{p}\mathit{l}\mathit{y}\mathit{ }\mathit{b}\mathit{o}\mathit{t}\mathit{h}\mathit{ }\mathit{s}\mathit{i}\mathit{d}\mathit{e}\mathit{ }\mathit{b}\mathit{y}\mathit{ }\frac{\mathit{d}}{\mathit{d}\mathit{x}}$

$$\begin{gathered} \text{ \\ dydx=ddx3x2+43 \\ } \end{gathered}$$

$\mathit{S}\mathit{t}\mathit{e}\mathit{p}\mathbf{ }\mathbf{2}\mathbf{:}\mathbf{ }\mathit{A}\mathit{p}\mathit{p}\mathit{l}\mathit{y}\mathbf{ }\mathit{t}\mathit{h}\mathit{e}\mathbf{ }\mathit{Q}\mathit{o}\mathit{u}\mathit{t}\mathit{i}\mathit{e}\mathit{n}\mathit{t}\mathbf{ }\mathit{R}\mathit{u}\mathit{l}\mathit{e}$

$$\begin{gathered} \text{ \\ ddxuv=vddx(u)-uddx(v)v2 \\ } \end{gathered}$$

$\mathit{G}\mathit{i}\mathit{v}\mathit{e}\mathit{n}\mathbf{:}$

$$\begin{gathered} \text{ \\ u=3} \end{gathered}$$

$$\begin{gathered} \text{ \\ dudx=0} \end{gathered}$$

$$\begin{gathered} \text{ \\ v=x2+43} \end{gathered}$$

$$\begin{gathered} \text{ \\ dvdx=6x(x2+4)2 \\ } \end{gathered}$$

$\mathit{S}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathbf{:}$

$$\begin{gathered} \text{ \\ dydx=(x2+4)0-36xx2+42x2+46} \end{gathered}$$

$$\begin{gathered} \text{ \\ dydx=-18xx2+42x2+46} \end{gathered}$$

$$\begin{gathered} \text{ \\ dydx=-18xx2+44 \\ } \end{gathered}$$

$\mathit{F}\mathit{i}\mathit{n}\mathit{a}\mathit{l}\mathbf{ }\mathit{A}\mathit{n}\mathit{s}\mathit{w}\mathit{e}\mathit{r}\mathbf{:} \frac{\mathbf{-}\mathbf{18}\mathit{x}}{{\left({\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\right)}^{\mathbf{4}}}$