AP Calculus AB and BC

Derivative of ln 3x: Answer, Proof, and Mistakes

The derivative of ln 3x is 1/x, valid for x > 0. In prime notation, if f(x) = ln 3x then f'(x) = 1/x. The chain rule gives 3/(3x), and the 3 cancels, so the answer is identical to the derivative of ln x. You can also see it from ln 3x = ln 3 + ln x, where ln 3 is a constant.

ddx[ln⁡3x]=1x\frac{d}{dx}\left[\ln 3x\right] = \frac{1}{x}

How to differentiate ln 3x

The outer function is the natural log and the inner is 3x3x. The chain rule form for a logarithm is ddxln⁡u=u′u\frac{d}{dx}\ln u = \frac{u'}{u}, so divide the inner derivative by the inner function.

ddxln⁡(3x)=33x=1x\frac{d}{dx}\ln(3x) = \frac{3}{3x} = \frac{1}{x}

The constant 33 appears on top and bottom and cancels. A cleaner route is to split the logarithm first, since ln⁡3x=ln⁡3+ln⁡x\ln 3x = \ln 3 + \ln x and ln⁡3\ln 3 is a constant.

ddx(ln⁡3+ln⁡x)=0+1x=1x\frac{d}{dx}\left(\ln 3 + \ln x\right) = 0 + \frac{1}{x} = \frac{1}{x}

Watch the domain

ln⁡3x\ln 3x requires 3x>03x > 0, that is x>0x > 0, so the derivative 1x\frac{1}{x} is stated on that domain.

Why the constant makes no difference

Any positive constant multiplier inside a log becomes an additive constant outside it, and constants have zero derivative. So ln⁡(cx)\ln(cx) has the same derivative 1x\frac{1}{x} as ln⁡x\ln x for every constant c>0c > 0.

This rule sits in Unit 2 (the derivative of ln⁡x\ln x, Topic 2.7) combined with the Unit 3 chain rule, Topic 3.1. It is a frequent way the exam checks whether you apply the chain rule mechanically or think about the structure first.

Common mistakes with the derivative of ln 3x

  • Answering 3x\frac{3}{x}, keeping the inner derivative 33 but forgetting to divide by the inner function 3x3x.
  • Answering 13x\frac{1}{3x}, dividing by the inside but forgetting the inner derivative 33 on top.
  • Assuming the 33 must survive somewhere. It genuinely cancels, and the final derivative is exactly 1x\frac{1}{x}.

Check yourself, not just the answer

Type derivatives and get graded on mathematical equivalence, with rule-level hints when you miss, in the Derivative Practice Checker.

Frequently asked questions

What is the derivative of ln⁡3x\ln 3x?

It is 1x\frac{1}{x}, the same as the derivative of ln⁡x\ln x, because the constant 33 cancels in the chain rule.

Why is it not 3x\frac{3}{x} or 13x\frac{1}{3x}?

The chain rule gives 33x\frac{3}{3x}, with the 33 on top from the inner derivative and 3x3x on the bottom from the inside. The threes cancel, leaving 1x\frac{1}{x}.

Does ln⁡(cx)\ln(cx) always have derivative 1x\frac{1}{x}?

Yes, for any constant c>0c > 0. Since ln⁡(cx)=ln⁡c+ln⁡x\ln(cx) = \ln c + \ln x and ln⁡c\ln c is constant, the derivative is 1x\frac{1}{x}.