\[\frac{d}{dx} \cos{(\ln{({x}^{3})})}\]
1
Use Chain Rule on ddxcos⁡(ln⁡(x3))\frac{d}{dx} \cos{(\ln{({x}^{3})})}. Let u=ln⁡(x3)u=\ln{({x}^{3})}. Use Trigonometric Differentiation: the derivative of cos⁡u\cos{u} is −sin⁡u-\sin{u}.
−sin⁡(ln⁡(x3))(ddxln⁡(x3))-\sin{(\ln{({x}^{3})})}(\frac{d}{dx} \ln{({x}^{3})})

2
Use Chain Rule on ddxln⁡(x3)\frac{d}{dx} \ln{({x}^{3})}. Let u=x3u={x}^{3}. The derivative of ln⁡u\ln{u} is 1u\frac{1}{u}.
−sin⁡(ln⁡(x3))×1x3(ddxx3)-\sin{(\ln{({x}^{3})})}\times \frac{1}{{x}^{3}}(\frac{d}{dx} {x}^{3})

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

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