derivative of (csc(x)/(2^x))

解答

\frac{d}{d x} (\frac{csc ( x )}{2 ^{x}})

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

求解步骤

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

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

\frac{d}{d x} (2^{x}) = 2^{x} ln (2)

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

化简

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

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

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

\frac{d}{d x} (- sin (5 x))

\int_{0}^{2 π} sin^{2} (8 x) d x

\frac{\partial}{\partial y} (27 x y^{2} + 8 y^{3})

x \to 2 lim (4 x - 5)