\[\frac{d}{dx} {(\cos{({x}^{3})})}^{2}\]
1
Remove parentheses.
ddxcos⁡2x3\frac{d}{dx} \cos^{2}{x}^{3}

2
Use Chain Rule on ddxcos⁡2x3\frac{d}{dx} \cos^{2}{x}^{3}. Let u=cos⁡(x3)u=\cos{({x}^{3})}. Use Power Rule: dduun=nun−1\frac{d}{du} {u}^{n}=n{u}^{n-1}.
2cos⁡(x3)(ddxcos⁡(x3))2\cos{({x}^{3})}(\frac{d}{dx} \cos{({x}^{3})})

3
Use Chain Rule on ddxcos⁡(x3)\frac{d}{dx} \cos{({x}^{3})}. Let u=x3u={x}^{3}. Use Trigonometric Differentiation: the derivative of cos⁡u\cos{u} is −sin⁡u-\sin{u}.
2cos⁡(x3)×−sin⁡(x3)(ddxx3)2\cos{({x}^{3})}\times -\sin{({x}^{3})}(\frac{d}{dx} {x}^{3})

4
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
−6x2cos⁡(x3)sin⁡(x3)-6{x}^{2}\cos{({x}^{3})}\sin{({x}^{3})}

Done
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