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Popular >

derivative of (1+e^{-x}^{-1})

Solution

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

Solution

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

Solution steps

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

Apply the chain rule:

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

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

\frac{d}{d 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}}

Popular Examples

(dy)/(dx)=y(1-y)

\frac{d y}{d x} = y (1 - y)

limit as x approaches 4 of 5x^2-2x+3

x \to 4 lim (5 x^{2} - 2 x + 3)

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x \to 7 lim (\frac{( x ^{2} - 2 x - 35 )}{x - 7})

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x \to 0 lim (\frac{1 ^{x} - 9 ^{x}}{x})

(\partial)/(\partial x)(x^2+2y^2-x^2y)

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

Frequently Asked Questions (FAQ)

What is the derivative of (1+e^{-x}^{-1}) ?
The derivative of (1+e^{-x}^{-1}) is (e^{-x})/((1+e^{-x))^2}
What is the first derivative of (1+e^{-x}^{-1}) ?
The first derivative of (1+e^{-x}^{-1}) is (e^{-x})/((1+e^{-x))^2}