Question

How do you find the derivative of `y=ln(5x)`?

Answer 1

`y'=1/x`

Explanation

Suppose, `y=ln(b(x))`

then using Chain Rule,

`y'=1/(b(x))*(b(x))'`

Similarly following for the above function yields,

`y'=1/(5x)*(5x)'`

`y'=1/(5x)*5`

`y'=1/x`

Answer 2

A student comfortable with the natural logarithm function and its properties might think of this:

One could reason as follows:
`y=ln(5x)=ln(5)+ln(x)`.

But `ln(5)` is a constant, so its derivative is `0`.

Therefore, `(dy)/(dx) = d/(dx)(ln5+lnx)=d/(dx)(lnx)=1/x`.

Answer 3

Using the Chain Rule you'll get:

`y'=1/(5x)*5=1/x`

(Derive `ln` as it is and then multiply by the derivative of the argument, `5x`).