derivative of (cot(x)/(e^x))

解答

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

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

求解步骤

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

使用微分除法定则:

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

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

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

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

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

化简

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

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

d er i v a t i v e 2 e^{x}

\int t sin^{2} (t) d t

x \to 0 lim (3 x^{2} - 5)

\int \frac{- 1}{1 + e ^{x}} d x

x \to - 5 lim (2 (- \frac{3}{10}) (x) - 3)