# What is the derivative of ln(8x)?

Oct 16, 2015

$\frac{1}{x}$

#### Explanation:

Rule : $\frac{d}{\mathrm{dx}} \ln u \left(x\right) = \frac{1}{u} \cdot \frac{\mathrm{du}}{\mathrm{dx}}$

$\therefore \frac{d}{\mathrm{dx}} \left[\ln \left(8 x\right)\right] = \frac{1}{8 x} \cdot \frac{d}{\mathrm{dx}} \left(8 x\right)$

$= \frac{8}{8 x} = \frac{1}{x}$

Oct 16, 2015

$\frac{1}{x}$

#### Explanation:

The derivative of $\ln \left(n x\right)$ is equal to $\left(\frac{\left(n x\right) '}{n x}\right)$ or the derivative of the inside of the natural log divided by the inside of the natural log.

The derivative of 8x is just 8, so it would be:

$\frac{8}{8 x}$ which is just $\frac{1}{x}$

Hope this helped!

Oct 16, 2015

An alternative solution using the properties of logarithms.

#### Explanation:

Instead of using the chain rule, we can use the property of logarithms $\ln \left(a b\right) = \ln a + \ln b$ to rewrite:

$y = \ln \left(8 x\right) = \ln 8 + \ln x$

Now, since $\ln 8$ is some constant, its derivative is $0$, so we get"

$\frac{\mathrm{dy}}{\mathrm{dx}} = 0 + \frac{1}{x} = \frac{1}{x}$