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

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

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

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

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derivative of

\frac{1}{d + k e ^{d}}

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

Solution steps

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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