What is the d/(du)(u/(u+v)) ?

The d/(du)(u/(u+v)) is v/((u+v)^2)

What is the first d/(du)(u/(u+v)) ?

The first d/(du)(u/(u+v)) is v/((u+v)^2)

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\frac{d}{d u} (\frac{u}{u + v})

\frac{v}{( u + v ) ^{2}}

Solution steps

\frac{d}{d u} (\frac{u}{u + v})

Treat

v

as a constant

Apply the Quotient Rule:

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

= \frac{\frac{d u}{d u} ( u + v ) - \frac{d}{d u} ( u + v ) u}{( u + v ) ^{2}}

\frac{d u}{d u} = 1

\frac{d}{d u} (u + v) = 1

= \frac{1 \cdot ( u + v ) - 1 \cdot u}{( u + v ) ^{2}}

1 \cdot (u + v) - 1 \cdot u = v

= \frac{v}{( u + v ) ^{2}}