Utilizing the General Power Rule in Finding the Derivative

$\mathit{F}\mathit{i}\mathit{n}\mathit{d}\mathbf{ }\mathit{t}\mathit{h}\mathit{e}\mathbf{ }\mathit{D}\mathit{e}\mathit{r}\mathit{i}\mathit{v}\mathit{a}\mathit{t}\mathit{i}\mathit{v}\mathit{e}\mathbf{ }\mathit{o}\mathit{f}\mathbf{:} \mathit{y}\mathbf{=}{\mathit{x}}^{\mathbf{5}}\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{2}\mathit{x}{\mathbf{)}}^{\mathbf{7}}$

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

$$\begin{gathered} \text{ \\ dydx=ddxx5ddx1+2x)7 \\ } \end{gathered}$$

$\mathit{S}\mathit{t}\mathit{e}\mathit{p}\mathbf{ }\mathbf{2}\mathbf{:}\mathbf{ }\mathit{A}\mathit{p}\mathit{p}\mathit{l}\mathit{y}\mathbf{ }\mathit{t}\mathit{h}\mathit{e} \mathit{G}\mathit{e}\mathit{n}\mathit{e}\mathit{r}\mathit{a}\mathit{l}\mathbf{ }\mathit{P}\mathit{o}\mathit{w}\mathit{e}\mathit{r}\mathbf{ }\mathit{R}\mathit{u}\mathit{l}\mathit{e} \mathit{i}\mathit{n}\mathbf{ }\frac{\mathbf{d}}{\mathbf{dx}}\left(\mathbf{1}\mathbf{+}\mathbf{2}\mathit{x}{\mathbf{)}}^{\mathbf{7}}\right.$

$$\begin{gathered} \text{ \\ ddxcun=cnun-1dudx \\ } \end{gathered}$$

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

$$\begin{gathered} \text{ \\ u=1+2x} \end{gathered}$$

$$\begin{gathered} \text{ \\ dudx=2} \end{gathered}$$

$$\begin{gathered} \text{ \\ n=7} \end{gathered}$$

$$\begin{gathered} \text{ \\ c=1 \\ } \end{gathered}$$

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

$$\begin{gathered} \text{ \\ =(7)(1+2x)7-1(2)} \end{gathered}$$

$$\begin{gathered} \text{ \\ =14(1+2x)6} \end{gathered}$$

$$\begin{gathered} \text{ \\ ddxx5=5x4} \end{gathered}$$

$$\begin{gathered} \text{ \\ dydx=5x4(14(1+2x)6) \\ } \end{gathered}$$

$\mathbf{S}\mathbf{i}\mathbf{m}\mathbf{p}\mathbf{l}\mathbf{i}\mathbf{f}\mathbf{y}\mathbf{:}\mathbf{ }70{\mathrm{x}}^4(1+2\mathrm{x}{)}^6$

$\mathit{F}\mathit{i}\mathit{n}\mathit{a}\mathit{l}\mathbf{ }\mathit{A}\mathit{n}\mathit{s}\mathit{w}\mathit{e}\mathit{r}\mathbf{:}\mathbf{ }\mathbf{70}{\mathbf{x}}^{\mathbf{4}}\left(\mathbf{1}\mathbf{+}\mathbf{2}\mathbf{x}{\mathbf{)}}^{\mathbf{6}}\right.$