Derivative of ln(2x): Why It Is 1/x, With Examples

The derivative of \ln(2x) is \dfrac{1}{x}. This surprises many students: the 2 disappears, and the answer matches the derivative of ln x.

Proof 1: the chain rule

Let u = 2x, so u' = 2. The chain rule for logs says "derivative of the inside over the inside":

The 2 on top cancels the 2 on the bottom.

Proof 2: split the log first

The product rule for logarithms says \ln(ab) = \ln a + \ln b. So

Now \ln 2 \approx 0.693 is just a number, and the derivative of a constant is 0:

This second proof explains why the answer is the same as for \ln x.

Graph: two curves, same slopes

At any x, the two curves are the same vertical distance apart, so their tangent lines are parallel. At x = 2, for example, both have slope \frac{1}{2}.

When the constant does not cancel

The trick only works when the inside is a constant times x. Compare:

The last row works because \ln(2x^3) = \ln 2 + 3\ln x.

Worked examples

Example 1: a tangent line. At x = \frac{1}{2}, \ln(2x) = \ln 1 = 0 and the slope is \frac{1}{1/2} = 2. The tangent line is

Example 2: product rule. For f(x) = x\ln(2x), the product rule gives

Example 3: a square. For g(x) = \left[\ln(2x)\right]^2, the outer function is a square:

Common mistakes

1. Writing \frac{1}{2x}. You divided by the inside but forgot to multiply by its derivative, 2.
2. Writing \frac{2}{x}. You multiplied by 2 but did not keep the 2 in the denominator.
3. Cancelling in \ln(2x + 1). The 2 only cancels when the inside is exactly 2x.

See also the derivative of ln(x²), where a power instead of a multiple sits inside the log, and practise more with derivatives of exponential and logarithmic functions.