\[\frac{d}{dx} \frac{1}{\tan{x}}\]
1
Use Chain Rule on ddx1tan⁡x\frac{d}{dx} \frac{1}{\tan{x}}. Let u=tan⁡xu=\tan{x}. Use Power Rule: dduun=nun−1\frac{d}{du} {u}^{n}=n{u}^{n-1}.
−1tan⁡2x(ddxtan⁡x)-\frac{1}{\tan^{2}x}(\frac{d}{dx} \tan{x})

2
Use Trigonometric Differentiation: the derivative of tan⁡x\tan{x} is sec⁡2x\sec^{2}x.
−1−1tan⁡2x-1-\frac{1}{\tan^{2}x}

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