Derivative of ln 2x: Answer, Why the 2 Vanishes

How to differentiate ln 2x

By the chain rule, the derivative of $\ln u$ is $\frac{u'}{u}$. With $u = 2x$ the numerator is $2$ and the denominator is $2x$.

The twos cancel, so the answer is identical to the derivative of $\ln x$.

The log rule explanation

Splitting the logarithm first makes the result obvious without any cancellation.

$\ln 2$ is a constant, so it differentiates to zero and only $\frac{1}{x}$ survives. Geometrically, $\ln 2x$ is $\ln x$ shifted vertically, and a vertical shift never changes slopes.

The same argument shows $\frac{d}{dx}\ln(kx) = \frac{1}{x}$ for ANY positive constant $k$.

Where the derivative of ln 2x shows up on the AP exam

Topics 3.1 and 2.8 cover this on both exams. It is a favourite multiple choice item precisely because the obvious answer $\frac{2}{x}$ is wrong and looks right.

Contrast it with $\ln\left(x^{2}\right)$, where the constant is an exponent rather than a factor: there the derivative IS $\frac{2}{x}$.

Common mistakes with the derivative of ln 2x

• Answering $\frac{2}{x}$, keeping the chain rule factor but forgetting it cancels.
• Answering $\frac{1}{2x}$, forgetting the chain rule factor entirely.
• Assuming the graphs are identical because the derivatives are. They differ by the constant $\ln 2$, which is invisible to differentiation but not to the function.