What is the derivative of arctan(e^{-x}) ?

The derivative of arctan(e^{-x}) is -(e^{-x})/(e^{-2x)+1}

What is the first derivative of arctan(e^{-x}) ?

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

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\frac{d}{d x} (arctan (e^{- x}))

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

Solution steps

\frac{d}{d x} (arctan (e^{- x}))

Apply the chain rule:

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

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

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

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

Simplify

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

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

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