Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 34

Determine the derivative of the function $\displaystyle y = x \sin \frac{1}{x}$


$
\begin{equation}
\begin{aligned}
y' &= \frac{d}{dx} \left( x \sin \frac{1}{x} \right)\\
\\
y' &= x \cdot \frac{d}{dx} \left( \sin \frac{1}{x} \right) + \sin \frac{1}{x} \cdot \frac{d}{dx} (x)\\
\\
y' &= x \cos \frac{1}{x} \cdot \frac{d}{dx} \left( \frac{1}{x} \right) + \left( \sin \frac{1}{x} \right) (1)\\
\\
y' &= (x) \left( \cos \frac{1}{x} \right) \left( \frac{-1}{x^2} \right) + \sin \frac{1}{x}\\
\\
y' &= \left( \cos \frac{1}{x} \right) \left(\frac{-1}{x} \right) + \sin \frac{1}{x}\\
\\
y' &= \sin \frac{1}{x} - \frac{\cos \frac{1}{x}}{x}
\end{aligned}
\end{equation}
$