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

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

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(\partial)/(\partial x)(1/((1+e^{-x))})

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

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

Solution steps

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d}{d x} (lo g_{10} ((x - 3)^{2}))

\int \frac{4 x ^{2} + 1}{2 x + 5} d x

\int e^{- \frac{3}{2} t} d t

3 x^{2} + 4 x + 8

(x^{2} + 7)^{5}

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