Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 52

Determine the equation of the tangent line to the curve $y= \sin x + \sin^2 x$ at the point $(0, 0)$.

Solving for the slope


$
\begin{equation}
\begin{aligned}

y' = m =& \frac{d}{dx} (\sin x + \sin^2 x)
\\
\\
m =& \frac{d}{dx} (\sin x) \frac{d}{dx} (\sin x)^2
\\
\\
m =& \cos x + 2 \sin x \cdot \frac{d}{dx} (\sin x)
\\
\\
m =& \cos x + 2 \sin x \cos x
\qquad \qquad \text{Apply Double Angle Formula $(\sin 2x = 2 \sin x \cos x)$}
\\
\\
m =& \cos x + \sin 2 x
\\
\\
m =& \cos (0) + \sin 2(0)
\\
\\
m =& 1 + 0
\\
\\
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 - 0)
\\
\\
y =& x
\qquad \qquad \text{Equation of the tangent line at $(0,0)$}

\end{aligned}
\end{equation}
$