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 $m_T$


$\begin{equation} \begin{aligned} y - y_1 =& m (x - x_1) \\ \\ y - 1 =& 1 (x - 0) \\ \\ y -1 =& x && \\ \\ y =& x + 1 \end{aligned} \end{equation}$