What is the d/(dy)((ax+by)/(cx+dy)) ?

The d/(dy)((ax+by)/(cx+dy)) is (bcx-adx)/((cx+dy)^2)

What is the first d/(dy)((ax+by)/(cx+dy)) ?

The first d/(dy)((ax+by)/(cx+dy)) is (bcx-adx)/((cx+dy)^2)

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

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

Solution steps

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

Treat

a, b, c, d, x

as a constant

Apply the Quotient Rule:

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

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

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

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

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

Expand

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

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

What is the d/(dy)((ax+by)/(cx+dy)) ?
The d/(dy)((ax+by)/(cx+dy)) is (bcx-adx)/((cx+dy)^2)
What is the first d/(dy)((ax+by)/(cx+dy)) ?
The first d/(dy)((ax+by)/(cx+dy)) is (bcx-adx)/((cx+dy)^2)