$\frac{d\left(\cos ^{2} x\right)^{3}}{d x}$

Step 1 ：\(\frac{d}{dx} (\cos^2 x)^3 = 3(\cos^2 x)^2 \cdot \frac{d}{dx} (\cos^2 x)\)

Step 2 ：\(\frac{d}{dx} (\cos^2 x) = 2 \cos x \cdot (-\sin x)\)

Step 3 ：\(3(\cos^2 x)^2 \cdot \frac{d}{dx} (\cos^2 x) = 3(\cos^2 x)^2 \cdot (2 \cos x \cdot (-\sin x))\)

Step 4 ：\(\frac{d}{dx} (\cos^2 x)^3 = \boxed{-6 \cos^3 x \sin x}\)