What is the d/(ds)(s/(s^2+w^2)) ?

The d/(ds)(s/(s^2+w^2)) is (w^2-s^2)/((s^2+w^2)^2)

What is the first d/(ds)(s/(s^2+w^2)) ?

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

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\frac{d}{d s} (\frac{s}{s ^{2} + w ^{2}})

\frac{w ^{2} - s ^{2}}{( s ^{2} + w ^{2} ) ^{2}}

Solution steps

\frac{d}{d s} (\frac{s}{s ^{2} + w ^{2}})

Treat

w

as a constant

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d s}{d s} ( s ^{2} + w ^{2} ) - \frac{d}{d s} ( s ^{2} + w ^{2} ) s}{( s ^{2} + w ^{2} ) ^{2}}

\frac{d s}{d s} = 1

\frac{d}{d s} (s^{2} + w^{2}) = 2 s

= \frac{1 \cdot ( s ^{2} + w ^{2} ) - 2 ss}{( s ^{2} + w ^{2} ) ^{2}}

1 \cdot (s^{2} + w^{2}) - 2 ss = w^{2} - s^{2}

= \frac{w ^{2} - s ^{2}}{( s ^{2} + w ^{2} ) ^{2}}

What is the d/(ds)(s/(s^2+w^2)) ?
The d/(ds)(s/(s^2+w^2)) is (w^2-s^2)/((s^2+w^2)^2)
What is the first d/(ds)(s/(s^2+w^2)) ?
The first d/(ds)(s/(s^2+w^2)) is (w^2-s^2)/((s^2+w^2)^2)