What is the derivative of ((ax+b)/(cx+d)) ?

The derivative of ((ax+b)/(cx+d)) is (-bc+ad)/((cx+d)^2)

What is the first derivative of ((ax+b)/(cx+d)) ?

The first derivative of ((ax+b)/(cx+d)) is (-bc+ad)/((cx+d)^2)

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

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

Solution steps

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

Apply the Quotient Rule:

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

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

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

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

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

Expand

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

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

What is the derivative of ((ax+b)/(cx+d)) ?
The derivative of ((ax+b)/(cx+d)) is (-bc+ad)/((cx+d)^2)
What is the first derivative of ((ax+b)/(cx+d)) ?
The first derivative of ((ax+b)/(cx+d)) is (-bc+ad)/((cx+d)^2)