partial derivative of ln(1+e^{-rt})

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∂∂r (ln(1+e^−rt))=−e^−rtt1+e^−rt

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∂ ∂ r (ln(1+e^−rt))=−e^−rtt1+e^−rt

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∂ ∂ r (ln(1+e^−rt))

Treat t as a constant

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Apply the chain rule: 11+e^−rt ∂ ∂ r (1+e^−rt)
=11+e^−rt ∂ ∂ r (1+e^−rt)

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∂ ∂ r (1+e^−rt)=−te^−tr
=11+e^−rt (−te^−tr)

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Simplify 11+e^−rt (−te^−tr): −e^−rtt1+e^−rt
=−e^−rtt1+e^−rt