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

1 Answer
Sep 6, 2017

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