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

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

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

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

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

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

Solution steps

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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