What is the first d/(ds)(1/s) ?

The first d/(ds)(1/s) is -1/(s^2)

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d/(ds)(1/s)

\frac{d}{d s} (\frac{1}{s})

- \frac{1}{s ^{2}}

Solution steps

\frac{d}{d s} (\frac{1}{s})

Apply exponent rule:

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

= \frac{d}{d s} (s^{- 1})

Apply the Power Rule:

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

= - 1 \cdot s^{- 1 - 1}

Simplify

- 1 \cdot s^{- 1 - 1} : - \frac{1}{s ^{2}}

= - \frac{1}{s ^{2}}