Question

What is the derivative of y=ln(3x)?

Answer 1

`1/x`

Explanation:

Recall that the natural log of any product is equal to the sum of the natural log of the components; that is to say, `ln (a*b)=ln a + ln b` . Thus, `ln 3x = ln 3 + ln x`

Further, recall that for any constant contact, `(dc)/(dx) = 0. Ln 3` is, of course, a constant. Thus, `d/(dx) ln 3 = 0`.

Finally, the derivative of the natural log function `ln x` with respect to x, is `1/x`. With all of these in mind, we can find the desired derivative:

`d/(dx) ln (3x) = d/(dx) ln 3 + d/ (dx) ln x r= 0 + 1/x = 1/x`

As a bonus, note that for any `ln cx`, where c is any constant, then for all `cx >0`, `(df)/(dx) = 1/x`