Application of the Square Root of a Function Rule in Calculating the Derivative

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

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

$$\begin{gathered} \text{ \\ dydx=ddx2x2-4x+33 \\ } \end{gathered}$$

$\mathit{S}\mathit{t}\mathit{e}\mathit{p}\mathbf{ }\mathbf{2}\mathbf{:}\mathbf{ }\mathit{A}\mathit{p}\mathit{p}\mathit{l}\mathit{y}\mathbf{ }\mathit{t}\mathit{h}\mathit{e} \mathit{S}\mathit{q}\mathit{u}\mathit{a}\mathit{r}\mathit{e}\mathbf{ }\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathbf{ }\mathit{o}\mathit{f}\mathbf{ }\mathit{a}\mathbf{ }\mathit{F}\mathit{u}\mathit{n}\mathit{c}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathbf{ }\mathit{R}\mathit{u}\mathit{l}\mathit{e} \mathit{i}\mathit{n}\mathbf{ }\frac{\mathrm{d}}{\mathrm{dx}}\sqrt[3]{2x^2-4x+3}$

$$\begin{gathered} \text{ \\ ddxu=12ududx \\ } \end{gathered}$$

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

$$\begin{gathered} \text{ \\ u=2x2-4x+3} \end{gathered}$$

$$\begin{gathered} \text{ \\ dudx=4x-4 \\ } \end{gathered}$$

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

$$\begin{gathered} \text{ \\ dydx=132x2-4x+3234x-4 \\ } \end{gathered}$$

$\mathit{F}\mathit{i}\mathit{n}\mathit{a}\mathit{l}\mathbf{ }\mathit{A}\mathit{n}\mathit{s}\mathit{w}\mathit{e}\mathit{r}\mathbf{:}\mathbf{ }\frac{\mathbf{4}\mathit{x}\mathbf{-}\mathbf{4}}{\mathbf{3}\sqrt[\mathbf{3}]{{\left(\mathbf{2}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{4}\mathit{x}\mathbf{+}\mathbf{3}\right)}^{\mathbf{2}}}}$