Application of Product Rule in getting the Derivative

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

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

$$\begin{gathered} \text{ \\ dydx=ddxx3-23x2+4 \\ } \end{gathered}$$

$\mathit{S}\mathit{t}\mathit{e}\mathit{p}\mathit{ }\mathit{2}\mathit{:}\mathit{ }\mathit{A}\mathit{p}\mathit{p}\mathit{l}\mathit{y}\mathit{ }\mathit{t}\mathit{h}\mathit{e}\mathit{ }\mathit{P}\mathit{r}\mathit{o}\mathit{d}\mathit{u}\mathit{c}\mathit{t}\mathit{ }\mathit{R}\mathit{u}\mathit{l}\mathit{e}$

$$\begin{gathered} \text{ \\ ddx(uv)=udvdx+vdudx \\ } \end{gathered}$$

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

$$\begin{gathered} \text{ \\ u = x3-2} \end{gathered}$$

$$\begin{gathered} \text{ \\ dudx=3x2} \end{gathered}$$

$$\begin{gathered} \text{ \\ v=3x2+4} \end{gathered}$$

$$\begin{gathered} \text{ \\ dvdx=3x3x2+4 \\ } \end{gathered}$$

$$\begin{gathered} \text{ \\ Solution:} \end{gathered}$$

$$\begin{gathered} \text{ \\ dydx=x3-23x3x2+4+3x2+4(3x2)} \end{gathered}$$

$$\begin{gathered} \text{ \\ dydx=3x4-6x3x2+4+3x23x2+4} \end{gathered}$$

$$\begin{gathered} \text{ \\ Simplify:} \end{gathered}$$

$$\begin{gathered} \text{ \\ dydx=12x4+12x2-6x3x2+4 \\ } \end{gathered}$$

$\mathit{F}\mathit{i}\mathit{n}\mathit{a}\mathit{l}\mathit{ }\mathit{A}\mathit{n}\mathit{s}\mathit{w}\mathit{e}\mathit{r}\mathit{:}\mathit{ }\frac{\mathit{12}{\mathit{x}}^{\mathbf{4}}\mathbf{+}\mathbf{12}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{6}\mathit{x}}{\sqrt{\mathbf{3}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}}}$