What is the derivative of 1/(1+ce^{-x}) ?

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

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

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

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

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

Solution steps

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

t \frac{d y}{d t} + 30 y = \frac{1}{t}

\int x e^{\frac{x}{2}} d x

(8.4) (5.2)

\int - \frac{3000}{x ^{2}} d x

\int \frac{10 x}{4 x ^{2} - 4} d x

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