What is the derivative of 1/(c+ke^c) ?

The derivative of 1/(c+ke^c) is -(e^c)/((c+ke^c)^2)

What is the first derivative of 1/(c+ke^c) ?

The first derivative of 1/(c+ke^c) is -(e^c)/((c+ke^c)^2)

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derivative

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

Solution steps

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

\frac{d}{d k} (c + k e^{c}) = e^{c}

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

Simplify

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

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

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