Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 53

Determine the equation of the tangent line to the curve $y= \sin (\sin x)$
at the point $(\pi, 0)$.

Solving for the slope


$\begin{equation} \begin{aligned} y' = m =& \frac{d}{dx} [\sin (\sin x)] \\ \\ m =& \cos (\sin x) \cdot \frac{d}{dx} ( \sin x) \\ \\ m =& \cos (\sin x ) \cos x \\ \\ m =& [\cos (\sin \pi)] (\cos \pi) \\ \\ m =& (1)(-1) \\ \\ m =& - 1 \end{aligned} \end{equation}$



Using the Point Slope Form



$\begin{equation} \begin{aligned} y - y_1 =& m (x - x_1) \\ \\ y - 0 =& -1 (x - \pi) \\ \\ y =& -x + \pi \qquad \qquad \text{Equation of the tangent line at $(\pi,0)$} \end{aligned} \end{equation}$