What is the derivative of ln((a-bx)/(a-cx)) ?

The derivative of ln((a-bx)/(a-cx)) is (-ab+ac)/((-bx+a)(-cx+a))

What is the first derivative of ln((a-bx)/(a-cx)) ?

The first derivative of ln((a-bx)/(a-cx)) is (-ab+ac)/((-bx+a)(-cx+a))

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derivative of

ln (\frac{a - b x}{a - c x})

\frac{- ab + a c}{( - b x + a ) ( - c x + a )}

Solution steps

\frac{d}{d x} (ln (\frac{a - b x}{a - c x}))

Apply the chain rule:

\frac{1}{\frac{a - b x}{a - c x}} \frac{d}{d x} (\frac{a - b x}{a - c x})

= \frac{1}{\frac{a - b x}{a - c x}} \frac{d}{d x} (\frac{a - b x}{a - c x})

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

= \frac{1}{\frac{a - b x}{a - c x}} \cdot \frac{- ab + a c}{( a - c x ) ^{2}}

Simplify

\frac{1}{\frac{a - b x}{a - c x}} \cdot \frac{- ab + a c}{( a - c x ) ^{2}} : \frac{- ab + a c}{( - b x + a ) ( - c x + a )}

= \frac{- ab + a c}{( - b x + a ) ( - c x + a )}

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\int \frac{1}{y ^{2} y ^{2} + 16} d y

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What is the derivative of ln((a-bx)/(a-cx)) ?
The derivative of ln((a-bx)/(a-cx)) is (-ab+ac)/((-bx+a)(-cx+a))
What is the first derivative of ln((a-bx)/(a-cx)) ?
The first derivative of ln((a-bx)/(a-cx)) is (-ab+ac)/((-bx+a)(-cx+a))