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

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

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

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

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

Solution steps

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

Treat

w, T

as constants

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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