Single Variable Calculus, Chapter 7, 7.3-2, Section 7.3-2, Problem 50

Determine the equation of the tangent line to the curve $\displaystyle y = \frac{e^x}{x}$ at the point $(1,e)$

Solving for the slope


$
\begin{equation}
\begin{aligned}

y' =& \frac{d}{dx} \left( \frac{e^x}{x} \right)
\\
\\
y' =& \frac{\displaystyle x \frac{d}{dx} (e^x) - (e^x) \frac{d}{dx} (x) }{x^2}
\\
\\
y' =& \frac{xe^x - e^x}{x^2}
\\
\\
y' =& \frac{e^x (x - 1)}{x^2}
\\
\\
y' =& \frac{e^1 (1 - 1)}{(1)^2}
\\
\\
y' =& \frac{e^1 (0)}{1}
\\
\\
y' =& \frac{0}{1}
\\
\\
y' =& 0

\end{aligned}
\end{equation}
$



Using Point Slope Form


$
\begin{equation}
\begin{aligned}

y - y_1 =& m(x - x_1)
\\
\\
y - e =& 0 (x - 1)
\\
\\
y - e =& 0
\\
\\
y =& e

\end{aligned}
\end{equation}
$


Equation of the tangent line at $(1, e)$