Tangent Line - Problem Solving

$\mathit{y}\mathbf{=}\sqrt{{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{25}}\mathbf{ }\mathit{a}\mathit{t}\mathbf{ }\mathit{t}\mathit{h}\mathit{e}\mathbf{ }\mathit{p}\mathit{o}\mathit{i}\mathit{n}\mathit{t}\mathbf{ }\mathbf{(}\mathbf{13}\mathbf{,}\mathbf{ }\mathbf{12}\mathbf{)}\mathbf{ }$

$\mathit{S}\mathit{t}\mathit{e}\mathit{p}\mathbf{ }\mathbf{1}\mathbf{:}\mathbf{ }\mathit{G}\mathit{e}\mathit{t}\mathbf{ }\mathit{t}\mathit{h}\mathit{e}\mathbf{ }\mathit{d}\mathit{e}\mathit{r}\mathit{i}\mathit{v}\mathit{a}\mathit{t}\mathit{i}\mathit{v}\mathit{e}\mathbf{ }\mathit{o}\mathit{f}\mathbf{ }\mathit{y}\mathbf{=}\sqrt{{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{25}}$
$Apply the Derivative of Square Root Rule$

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

$$\begin{gathered} \text{ \\ dydx=12x2-25(2x)} \end{gathered}$$

$$\begin{gathered} \text{ \\ dydx=xx2-25 \\ } \end{gathered}$$

$\mathit{S}\mathit{t}\mathit{e}\mathit{p}\mathit{ }\mathit{2}\mathit{:}\mathit{ }\mathit{G}\mathit{e}\mathit{t}\mathit{ }\mathit{t}\mathit{h}\mathit{e}\mathit{ }\mathit{s}\mathit{l}\mathit{o}\mathit{p}\mathit{e}\mathit{ }\mathit{a}\mathit{t}\mathit{ }\mathit{p}\mathit{o}\mathit{i}\mathit{n}\mathit{t}\mathit{ }\mathit{(}\mathit{13}\mathit{,}\mathit{ }\mathit{12}\mathit{)}$

$$\begin{gathered} \text{ \\ m=13132-25} \end{gathered}$$

$$\begin{gathered} \text{ \\ m=1312 \\ } \end{gathered}$$

$\mathit{S}\mathit{t}\mathit{e}\mathit{p}\mathbf{ }\mathbf{3}\mathbf{:}\mathbf{ }\mathit{U}\mathbf{s}\mathbf{i}\mathbf{n}\mathit{g}\mathbf{ }\mathit{t}\mathit{h}\mathit{e}\mathbf{ }\mathit{p}\mathit{o}\mathit{i}\mathit{n}\mathit{t}\mathbf{-}\mathit{s}\mathit{l}\mathit{o}\mathit{p}\mathit{e}\mathbf{ }\mathit{f}\mathit{o}\mathit{r}\mathit{m}\mathbf{ }\mathit{a}\mathit{t}\mathbf{ }\mathit{p}\mathit{o}\mathit{i}\mathit{n}\mathit{t}\mathbf{ }\mathbf{(}\mathbf{13}\mathbf{,}\mathbf{ }\mathbf{12}\mathbf{)}$

$$\begin{gathered} \text{ \\ y-12=1312(x-13)} \end{gathered}$$

$$\begin{gathered} \text{ \\ y-12=1312(x-13)12 \\ 12} \end{gathered}$$

$$\begin{gathered} \text{ \\ y-144=13(x-13) \\ 12} \end{gathered}$$

$$\begin{gathered} \text{ \\ y-144-13x+169=0} \end{gathered}$$

$$\begin{gathered} \text{ \\ 12y-13x+25=0} \end{gathered}$$

$$\begin{gathered} \text{ \\ 13x-12y-25=0 \\ } \end{gathered}$$

$\mathit{F}\mathit{i}\mathit{n}\mathit{a}\mathit{l}\mathbf{ }\mathit{A}\mathit{n}\mathit{s}\mathit{w}\mathit{e}\mathit{r}\mathbf{:}\mathbf{ }\mathbf{13}\mathit{x}\mathbf{-}\mathbf{12}\mathit{y}\mathbf{-}\mathbf{25}\mathbf{=}\mathbf{0}$