# Find the derivative of the function. y=ln(frac{4x}{x-6})

Find the derivative of the function. $y=\mathrm{ln}\left(\frac{4x}{x-6}\right)$
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Asma Vang
Apply Logarithmic Properties $\mathrm{ln}\left(\frac{a}{b}\right)=\mathrm{ln}a-\mathrm{ln}b$, so
$y=\mathrm{ln}\left(\frac{4x}{x-6}\right)=\mathrm{ln}\left(4x\right)-\mathrm{ln}\left(x-6\right)$
Differentiate both sides with respect to x
${y}^{\prime }=\frac{d}{dx}\left[\mathrm{ln}\left(4x\right)\right]-\frac{d}{dx}\left[\mathrm{ln}\left(x-6\right)\right]$
Apply $\frac{d}{dx}\left[\mathrm{ln}u\right]=\frac{{u}^{\prime }}{{u}^{\prime }}$ thus
${y}^{\prime }=\frac{4}{4x}-\frac{1}{x-6}$
Simplify
${y}^{\prime }=\frac{1}{x}-\frac{1}{x-6}$
Apply difference rule for two fractions $\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}$, to obtain
${y}^{\prime }=\frac{x-6-x}{x\left(x-6\right)}=\frac{-6}{{x}^{2}-6x}$
${y}^{\prime }=\frac{-6}{{x}^{2}-6x}$
Result
${y}^{\prime }=\frac{-6}{{x}^{2}-6x}$