How do you differentiate f(x)=lnx * sin4x using the product rule?

$\frac{\sin 4 x}{x} + 4 \ln x \cos 4 x$
$\left(\ln x\right) ' \left(\sin 4 x\right) + \left(\sin 4 x\right) ' \ln x$
$= \left(\ln x\right) = \frac{1}{x}$
$= \left(\sin 4 x\right) ' = \left(\cos 4 x\right) \left(4 x\right) ' = 4 \cos 4 x$
$= \frac{1}{x} \sin 4 x + 4 \cos 4 x \ln x$
$= \frac{\sin 4 x}{x} + 4 \ln x \cos 4 x$