What is the derivative of ln((4x/(x+9))) ?

The derivative of ln((4x/(x+9))) is 9/(x(x+9))

What is the first derivative of ln((4x/(x+9))) ?

The first derivative of ln((4x/(x+9))) is 9/(x(x+9))

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\frac{d}{d x} (ln (\frac{4 x}{x + 9}))

\frac{9}{x ( x + 9 )}

Solution steps

\frac{d}{d x} (ln (\frac{4 x}{x + 9}))

Apply the chain rule:

\frac{1}{\frac{4 x}{x + 9}} \frac{d}{d x} (\frac{4 x}{x + 9})

= \frac{1}{\frac{4 x}{x + 9}} \frac{d}{d x} (\frac{4 x}{x + 9})

\frac{d}{d x} (\frac{4 x}{x + 9}) = \frac{36}{( x + 9 ) ^{2}}

= \frac{1}{\frac{4 x}{x + 9}} \cdot \frac{36}{( x + 9 ) ^{2}}

Simplify

\frac{1}{\frac{4 x}{x + 9}} \cdot \frac{36}{( x + 9 ) ^{2}} : \frac{9}{x ( x + 9 )}

= \frac{9}{x ( x + 9 )}

What is the derivative of ln((4x/(x+9))) ?
The derivative of ln((4x/(x+9))) is 9/(x(x+9))
What is the first derivative of ln((4x/(x+9))) ?
The first derivative of ln((4x/(x+9))) is 9/(x(x+9))