What is (\partial)/(\partial x)(((ax+by))/((cx+dy))) ?

The answer to (\partial)/(\partial x)(((ax+by))/((cx+dy))) is (-byc+ayd)/((xc+yd)^2)

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(\partial)/(\partial x)(((ax+by))/((cx+dy)))

\frac{\partial}{\partial x} (\frac{( a x + b y )}{( c x + d y )})

\frac{- b yc + a y d}{( x c + y d ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{( a x + b y )}{( c x + d y )})

Treat

a, b, y, c, d

as constants

Apply the Quotient Rule:

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

= \frac{\frac{\partial}{\partial x} ( a x + b y ) ( c x + d y ) - \frac{\partial}{\partial x} ( c x + d y ) ( a x + b y )}{( c x + d y ) ^{2}}

\frac{\partial}{\partial x} (a x + b y) = a

\frac{\partial}{\partial x} (c x + d y) = c

= \frac{a ( c x + d y ) - c ( a x + b y )}{( c x + d y ) ^{2}}

Expand

a (c x + d y) - c (a x + b y) : - b yc + a y d

= \frac{- b yc + a y d}{( x c + y d ) ^{2}}

\frac{( 3 x + 5 ) ( 2 x - 7 )}{8 x ^{2} - 3}

\frac{\partial}{\partial x} (x y - 3 f (x, y) - f (x, y) x^{2} + f (x, y) y)

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What is (\partial)/(\partial x)(((ax+by))/((cx+dy))) ?
The answer to (\partial)/(\partial x)(((ax+by))/((cx+dy))) is (-byc+ayd)/((xc+yd)^2)