Solved: "Find the derivative of the function. y=ln⁡(4xx−6)"

Bevan Mcdonald

Answered question

2021-01-04

Find the derivative of the function.

y = \ln (\frac{4 x}{x - 6})

Asma Vang

Skilled2021-01-05Added 93 answers

Apply Logarithmic Properties

\ln (\frac{a}{b}) = \ln a - \ln b

, so

y = \ln (\frac{4 x}{x - 6}) = \ln (4 x) - \ln (x - 6)

Differentiate both sides with respect to x

y^{'} = \frac{d}{d x} [\ln (4 x)] - \frac{d}{d x} [\ln (x - 6)]

Apply

\frac{d}{d x} [\ln u] = \frac{u^{'}}{u^{'}}

thus

y^{'} = \frac{4}{4 x} - \frac{1}{x - 6}

Simplify

y^{'} = \frac{1}{x} - \frac{1}{x - 6}

Apply difference rule for two fractions

\frac{a}{b} - \frac{c}{d} = \frac{a d - b c}{b d}

, to obtain

y^{'} = \frac{x - 6 - x}{x (x - 6)} = \frac{- 6}{x^{2} - 6 x}

y^{'} = \frac{- 6}{x^{2} - 6 x}

Result

y^{'} = \frac{- 6}{x^{2} - 6 x}

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