What is the derivative of  ln(3x)?

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15
May 9, 2017

Derivative of ln(3x) is $\frac{1}{3} x \cdot 3$ or $\frac{1}{x}$ because both 3s
cancel each other.

Explanation:

$\frac{d}{\mathrm{dx}} \ln \left(3 x\right)$ = $\frac{1}{3} x \frac{d}{\mathrm{dx}} \left(3 x\right)$
$\frac{d}{\mathrm{dx}} \ln \left(3 x\right)$ = $\frac{1}{3} x \cdot 3$
$\frac{d}{\mathrm{dx}} \ln \left(3 x\right)$ = $\frac{1}{x}$

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5
Oct 3, 2016

$\ln \left(3 x\right) = y$

${e}^{y} = 3 x$

Now use implicit differentiation. Remember that:

$\frac{\mathrm{dy}}{\mathrm{dy}} \cdot \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{dx}}$

If you use implicit differentiation...

${e}^{y} = 3 x$

Should transform into...

${e}^{y} \cdot \frac{\mathrm{dy}}{\mathrm{dx}} = 3$

Therefore:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3}{e} ^ y$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3}{3 x}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{x}$

You could also differentiate it like this...

$y = \ln \left(3 x\right)$

$y = \ln \left(3\right) + \ln \left(x\right)$

*Because of logarithmic rules.

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{x}$

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