Single Variable Calculus, Chapter 3, 3.4, Section 3.4, Problem 22

Determine the equation of the tangent line to the curve $y = (1 + x) \cos x$ at the given point $(0,1)$


$
\begin{equation}
\begin{aligned}

\qquad y' =& (1 + x) \frac{d}{dx} (\cos x) + \cos x \frac{d}{dx} (1 + x)
&& \text{Using Product Rule}
\\
\\
\qquad y' =& (1 + x) (- \sin x) ++ (\cos x) (1)
&& \text{Simplify the equation}
\\
\\
\qquad y' =& - \sin x - x \sin x + \cos x
&&

\end{aligned}
\end{equation}
$


Let $y' = m_T$ (slope of the tangent line)


$
\begin{equation}
\begin{aligned}

y' = m_T =& - \sin (0) - (0) (\sin 0) + \cos (0)
&& \text{Substitute value of $x$}
\\
\\
m_T =& 1
&&

\end{aligned}
\end{equation}
$


Using Point Slope Form substitute the values of $x, y$ and $m_T$


$
\begin{equation}
\begin{aligned}

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

\end{aligned}
\end{equation}
$