derivative of (e^x/(e^x+5))

解答

\frac{d}{d x} (\frac{e ^{x}}{e ^{x} + 5})

\frac{5 e ^{x}}{( e ^{x} + 5 ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{e ^{x}}{e ^{x} + 5})

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} ( e ^{x} ) ( e ^{x} + 5 ) - \frac{d}{d x} ( e ^{x} + 5 ) e ^{x}}{( e ^{x} + 5 ) ^{2}}

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

\frac{d}{d x} (e^{x} + 5) = e^{x}

= \frac{e ^{x} ( e ^{x} + 5 ) - e ^{x} e ^{x}}{( e ^{x} + 5 ) ^{2}}

化简

\frac{e ^{x} ( e ^{x} + 5 ) - e ^{x} e ^{x}}{( e ^{x} + 5 ) ^{2}} : \frac{5 e ^{x}}{( e ^{x} + 5 ) ^{2}}

= \frac{5 e ^{x}}{( e ^{x} + 5 ) ^{2}}

\int 3 cos^{2} (x) sin (2 x) d x

\frac{d}{d x} (8 4 x - 1)

x \to 2 lim (x^{2} + a)

y^{^{'''}} - 3 y^{^{''}} + 3 y^{^{'}} - y = x

\int \frac{x}{2} + \frac{2}{x} d x