Question

Differentiate log[e^3x (5x-3/4x+2) ^1/3]

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Answer 1

Answer:

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Answer 2

 $\frac{dy}{dx}=3+\frac{22}{3(5x-3)(4x+2)}$

Given,

Let $y=\log(e^{3x}(\frac{5x-3}{4x+2} )^\frac{1}{3} )$

⇒ $y=\log e^{3x} + \log(\frac{5x-3}{4x+2})^{\frac{1}{3} }$$y=3x\log e+\frac{1}{3} \log\frac{5x-3}{4x+2}$

⇒ $y=3x+\frac{1}{3} \log\frac{5x-3}{4x+2}$

⇒ $y=3x+\frac{1}{3}(\log(5x-3)-\log(4x+2))$

Now, differentiating $y$ with respect to $x$,

$\frac{dy}{dx} = 3+\frac{1}{3} (\frac{5}{5x-3} -\frac{4}{4x+2})$

⇒ $\frac{dy}{dx} = 3 + \frac{1}{3} (\frac{20x+10-20x+12}{(5x-3)(4x+2)})$

∴  $\frac{dy}{dx}=3+\frac{22}{3(5x-3)(4x+2)}$

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