Single Variable Calculus, Chapter 3, 3.4, Section 3.4, Problem 23

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


$
\begin{equation}
\begin{aligned}

\qquad y' =& \frac{d}{dx} (x) + \frac{d}{dx} (\cos x)
&&
\\
\\
\qquad y' =& 1- \sin x
&&

\end{aligned}
\end{equation}
$




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




$
\begin{equation}
\begin{aligned}

y' = m_T =& 1 - \sin (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}
$