Question

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

Answer 1

`1/x`

Explanation:

Rule : `d/dxlnu(x)=1/u*(du)/dx`

`therefore d/dx[ln(8x)]=1/(8x)*d/dx(8x)`

`=8/(8x)=1/x`

Answer 2

`1/x`

Explanation:

The derivative of `ln(nx)` is equal to `(((nx)')/(nx))` 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:

`8/(8x)` which is just `1/x`

Hope this helped!

Answer 3

An alternative solution using the properties of logarithms.

Explanation:

Instead of using the chain rule, we can use the property of logarithms `ln(ab) = lna+lnb` to rewrite:

`y = ln(8x) = ln8+lnx`

Now, since `ln8` is some constant, its derivative is `0`, so we get"

`dy/dx = 0+1/x=1/x`