What is the derivative of 1/(e^x-1) ?

The derivative of 1/(e^x-1) is -(e^x)/((e^x-1)^2)

What is the first derivative of 1/(e^x-1) ?

The first derivative of 1/(e^x-1) is -(e^x)/((e^x-1)^2)

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

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

Solution steps

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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