Is the derivative of ln(ax) = 1/x ?

Question

Since it would be 1/(ax) times a

so that'd make 1/x right?

Answer 1

absolutely,

d/dx[ln(ax)]
1/ax * d/dx[ax]
1/ax * a
1/x

Answer 2

another way of looking at this is:
ln(ax) = ln(a) + ln(x)
ln(a) is a constant and its derivative is hence 0

so when you take the derivative, only the ln(x) term counts.

sometimes you get useful results when you do the reverse:
integral 1/x is ln(abs(x)) + c
there are a few instances when moving the c into the ln gives you better expressions, especially if you need to take e to the on both sides.

Answer 3

Yes. An easier way is to right ln ax=ln x + ln a
ln a is a constant.
d/dx ln x=1/x
d/dx ln a=0 so d/dx ln ax=1/x

Answer 4

Yes, you are correct.