What is (\partial)/(\partial p)((p+q)/(p+r)) ?

The answer to (\partial)/(\partial p)((p+q)/(p+r)) is (-q+r)/((p+r)^2)

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(\partial)/(\partial p)((p+q)/(p+r))

\frac{\partial}{\partial p} (\frac{p + q}{p + r})

\frac{- q + r}{( p + r ) ^{2}}

Solution steps

\frac{\partial}{\partial p} (\frac{p + q}{p + r})

Treat

q, r

as constants

Apply the Quotient Rule:

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

= \frac{\frac{\partial}{\partial p} ( p + q ) ( p + r ) - \frac{\partial}{\partial p} ( p + r ) ( p + q )}{( p + r ) ^{2}}

\frac{\partial}{\partial p} (p + q) = 1

\frac{\partial}{\partial p} (p + r) = 1

= \frac{1 \cdot ( p + r ) - 1 \cdot ( p + q )}{( p + r ) ^{2}}

Simplify

\frac{1 \cdot ( p + r ) - 1 \cdot ( p + q )}{( p + r ) ^{2}} : \frac{- q + r}{( p + r ) ^{2}}

= \frac{- q + r}{( p + r ) ^{2}}

What is (\partial)/(\partial p)((p+q)/(p+r)) ?
The answer to (\partial)/(\partial p)((p+q)/(p+r)) is (-q+r)/((p+r)^2)