Combination of Power of x Rule and Quotient Rule in Getting the DERIVATIVE

$Find the Derivative\mathit{ }of\mathit{:}\mathit{ }\mathit{y}\mathit{=}{\left(\frac{{\mathit{x}}^{\mathit{3}}\mathit{-}\mathit{2}}{{\mathit{x}}^{\mathit{3}}\mathbf{+}\mathbf{1}}\right)}^{\mathit{3}}$

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

$$\begin{gathered} \text{ \\ dydx=ddxx3-2x3+13 \\ } \end{gathered}$$

$Step\mathit{ }\mathit{2}\mathit{:}\mathit{ }\mathit{A}\mathit{p}\mathit{p}\mathit{l}\mathit{y}\mathit{ }\mathit{t}\mathit{h}\mathit{e}\mathit{ }\mathit{p}\mathit{o}\mathit{w}\mathit{e}\mathit{r}\mathit{ }\mathit{o}\mathit{f}\mathit{ }\mathit{x}$

$$\begin{gathered} \text{ \\ ddx(cxn)=cnxn-1 \\ } \end{gathered}$$

$\frac{dy}{dx}\mathit{=}\mathit{3}{\left(\frac{x^3-2}{x^3+1}\right)}^2$

$$\begin{gathered} \text{ \\ dydx=(3)x3-22x3+12 \\ } \end{gathered}$$

$Step\mathit{ 3}\mathit{:}\mathit{ }\mathit{A}\mathit{p}\mathit{p}\mathit{l}\mathit{y}\mathit{ }\mathit{t}\mathit{h}\mathit{e}\mathit{ }\mathit{Q}\mathit{u}\mathit{o}\mathit{t}\mathit{i}\mathit{e}\mathit{n}\mathit{t}\mathit{ }\mathit{R}\mathit{u}\mathit{l}\mathit{e}$

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

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

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

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

$$\begin{gathered} \text{ \\ v=x3+1} \end{gathered}$$

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

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

$$\begin{gathered} \text{ \\ dydx=(3)3x2x3+1-3x2x3-2(x3+1)22 \\ } \end{gathered}$$

$\mathit{F}\mathit{i}\mathit{n}\mathit{a}\mathit{l}\mathbf{ }\mathit{A}\mathit{n}\mathit{s}\mathit{w}\mathit{e}\mathit{r}\mathbf{:}\mathbf{ }\frac{\mathbf{27}{\mathit{x}}^{\mathbf{2}}\mathbf{(}{\mathit{x}}^{\mathbf{3}}\mathbf{-}\mathbf{2}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{(}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{1}{\mathbf{)}}^{\mathbf{4}}}$