Differentiate the following with respect to $x$: $\cos x \cdot \cos 2x \cdot \cos 3x$

Let $y = \cos x \cdot \cos 2x \cdot \cos 3x$.
Taking the logarithm on both sides,we obtain:
$\log y = \log (\cos x \cdot \cos 2x \cdot \cos 3x)$
$\Rightarrow \log y = \log (\cos x) + \log (\cos 2x) + \log (\cos 3x)$.
Differentiating both sides with respect to $x$,we obtain:
$\frac{1}{y} \frac{dy}{dx} = \frac{1}{\cos x} \cdot \frac{d}{dx}(\cos x) + \frac{1}{\cos 2x} \cdot \frac{d}{dx}(\cos 2x) + \frac{1}{\cos 3x} \cdot \frac{d}{dx}(\cos 3x)$.
$\Rightarrow \frac{1}{y} \frac{dy}{dx} = \frac{-\sin x}{\cos x} + \frac{-\sin 2x \cdot 2}{\cos 2x} + \frac{-\sin 3x \cdot 3}{\cos 3x}$.
$\Rightarrow \frac{dy}{dx} = y [-\tan x - 2\tan 2x - 3\tan 3x]$.
Substituting the value of $y$,we get:
$\frac{dy}{dx} = -\cos x \cdot \cos 2x \cdot \cos 3x [\tan x + 2\tan 2x + 3\tan 3x]$.