derivative (cot(t))/(e^t)

解答

导数

\frac{cot ( t )}{e ^{t}}

- \frac{csc ^{2} ( t ) + cot ( t )}{e ^{t}}

求解步骤

\frac{d}{d t} (\frac{cot ( t )}{e ^{t}})

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d t} ( cot ( t ) ) e ^{t} - \frac{d}{d t} ( e ^{t} ) cot ( t )}{( e ^{t} ) ^{2}}

\frac{d}{d t} (cot (t)) = - csc^{2} (t)

\frac{d}{d t} (e^{t}) = e^{t}

= \frac{( - csc ^{2} ( t ) ) e ^{t} - e ^{t} cot ( t )}{( e ^{t} ) ^{2}}

化简

\frac{( - csc ^{2} ( t ) ) e ^{t} - e ^{t} cot ( t )}{( e ^{t} ) ^{2}} : - \frac{csc ^{2} ( t ) + cot ( t )}{e ^{t}}

= - \frac{csc ^{2} ( t ) + cot ( t )}{e ^{t}}

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