Find the Slope of the given curve at the indicated point

$\mathit{Find\; the\; slope\; of\; the\; given\; curve\; at\; the\; indicated\; point.}$

$\mathit{y}\mathbf{=}{\left(\mathbf{4}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{3}\right)}^{\mathbf{2}} \mathit{;} \left(\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\mathbf{16}\right)$

$\mathit{S}\mathit{t}\mathit{e}\mathit{p}\mathbf{ }\mathbf{1}\mathbf{:}\mathbf{ }\mathit{U}\mathit{s}\mathit{e}\mathbf{ }\mathit{t}\mathit{h}\mathit{e}\mathbf{ }\mathit{P}\mathit{o}\mathit{w}\mathit{e}\mathit{r}\mathbf{ }\mathit{F}\mathit{o}\mathit{r}\mathit{m}\mathit{u}\mathit{l}\mathit{a}\mathbf{ }$

$\frac{\mathit{d}}{\mathit{d}\mathit{x}}\left(\mathit{c}{\mathit{u}}^{\mathit{n}}\right)\mathbf{=}\mathit{c}\mathit{n}{\mathit{u}}^{\mathit{n}\mathbf{-}\mathbf{1}}\frac{\mathit{d}}{\mathit{d}\mathit{x}}\left(\mathit{u}\right)\mathbf{ }\mathit{t}\mathit{o}\mathbf{ }\mathit{f}\mathit{i}\mathit{n}\mathit{d}\mathbf{ }{\mathit{f}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ 

$f^{\text{'}}\left(x\right)=\frac{dy}{dx}=2\left(4x^2+3\right)\left(8x\right)=16x\left(4x^2+3\right)$

$\mathit{N}\mathit{o}\mathit{t}\mathit{e}\mathbf{:}\mathbf{ }\mathit{A}\mathit{t}\mathbf{ }\mathit{p}\mathit{o}\mathit{i}\mathit{n}\mathit{t}\mathbf{ }\left(\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\mathbf{ }\mathbf{16}\right)\mathbf{,}\mathbf{ }\mathit{s}\mathit{l}\mathit{o}\mathit{p}\mathit{e}\mathbf{ }\mathit{o}\mathit{f}\mathbf{ }\mathbf{t}\mathbf{a}\mathbf{n}\mathit{g}\mathit{e}\mathit{n}\mathit{t}\mathbf{ }\mathit{l}\mathit{i}\mathit{n}\mathit{e}\mathbf{ }\mathit{i}\mathit{s}\mathbf{ }\mathit{g}\mathit{i}\mathit{v}\mathit{e}\mathit{n}\mathbf{ }\mathit{b}\mathit{e}\mathit{l}\mathit{o}\mathit{w}\mathbf{.}$

$$\begin{gathered} \text{ \\ f'12=16124122+3=8414+3=32} \end{gathered}$$

$\mathit{F}\mathit{i}\mathit{n}\mathit{a}\mathit{l}\mathbf{ }\mathit{A}\mathit{n}\mathit{s}\mathit{w}\mathit{e}\mathit{r}\mathbf{:}\mathbf{ }\mathit{T}\mathit{h}\mathit{e}\mathbf{ }\mathit{S}\mathit{l}\mathit{o}\mathit{p}\mathit{e}\mathbf{ }\mathit{o}\mathit{f}\mathbf{ }\mathit{t}\mathit{h}\mathit{e}\mathbf{ }\mathit{c}\mathit{u}\mathit{r}\mathit{v}\mathit{e}\mathbf{ }\mathit{a}\mathit{t}\mathbf{ }\left(\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\mathbf{ }\mathbf{16}\right)\mathbf{ }\mathit{i}\mathit{s}\mathbf{ }\mathbf{32}$