What is (\partial)/(\partial x)((e^x)/(1-e^x)) ?

The answer to (\partial)/(\partial x)((e^x)/(1-e^x)) is (e^x)/((1-e^x)^2)

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(\partial)/(\partial x)((e^x)/(1-e^x))

\frac{\partial}{\partial x} (\frac{e ^{x}}{1 - e ^{x}})

\frac{e ^{x}}{( 1 - e ^{x} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{e ^{x}}{1 - e ^{x}})

Apply the Quotient Rule:

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

= \frac{\frac{\partial}{\partial x} ( e ^{x} ) ( 1 - e ^{x} ) - \frac{\partial}{\partial x} ( 1 - e ^{x} ) e ^{x}}{( 1 - e ^{x} ) ^{2}}

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

\frac{\partial}{\partial x} (1 - e^{x}) = - e^{x}

= \frac{e ^{x} ( 1 - e ^{x} ) - ( - e ^{x} ) e ^{x}}{( 1 - e ^{x} ) ^{2}}

Simplify

\frac{e ^{x} ( 1 - e ^{x} ) - ( - e ^{x} ) e ^{x}}{( 1 - e ^{x} ) ^{2}} : \frac{e ^{x}}{( 1 - e ^{x} ) ^{2}}

= \frac{e ^{x}}{( 1 - e ^{x} ) ^{2}}

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What is (\partial)/(\partial x)((e^x)/(1-e^x)) ?
The answer to (\partial)/(\partial x)((e^x)/(1-e^x)) is (e^x)/((1-e^x)^2)