Derivative of ln(2x)

Answer

\frac{1}{x}

Derivative of ln(2x)

Rule Used: Chain Rule
Steps: Identify a composite function in $\log(2 \cdot x)$ : use the chain rule $\frac{d}{dx} f(g(x))$ = f'(g(x))·g'(x). The inner derivative is required because a stretch or shift of the input changes how fast the output moves.
Differentiate the outer function, then multiply by the inner derivative.
Final answer: $x^{-1}$

d/dx ln(2x) = x^(-1). The expression needs x > 0. At 0 and on the negatives it is undefined as a real function, so the derivative is too.

Instant step-by-step solution for ln(2x)

Derivative LogarithmicIntermediate

Share this result

Share on X

Derivative of ln(2x) = x^(-1) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-ln-2x

Derivative

ln(2x) → x^(-1)engine: x^(-1)

Graph of the function and its derivative

The solid curve is f. The dashed curve is f′. Vertical gaps are asymptotes or jumps, not extra zeros.

Why the derivative has this form

ln(2x) looks like it should carry a 2, but the log property says ln(2x) = ln 2 + ln x. The derivative of the constant ln 2 is 0, so the derivative is 1/x, the same as the derivative of ln x. The chain rule agrees: the derivative of ln(u) is 1/u, times u' = 2, and 2/(2x) cancels to 1/x. The 2 is real and then it cancels.

Apply the chain rule to ln(2x). The form to remember, already simplified, is x^(-1).

The expression needs x > 0. At 0 and on the negatives it is undefined as a real function, so the derivative is too.

At x = 2, substitute into the simplified form x^(-1). That number is the slope of ln(2x) there; a secant over a small step in x lands next to it.

Worked derivative

Write f(x) = ln(2x).
Apply the chain rule to ln(2x). The form to remember, already simplified, is x^(-1).
At x = 2, substitute into the simplified form x^(-1). That number is the slope of ln(2x) there; a secant over a small step in x lands next to it.

Common mistakes

Answering 1/(2*x)

1/(2x) stops before canceling. The chain rule really does multiply by 2, and that 2 cancels the 2 in the denominator.

Confusing the base of ln(2x)

Natural log differentiates to 1/x. A constant inside the log often cancels. Base 10 is a different function and needs log10.

Chain rule — The rule page lists this example and the others in the same family.
Logarithmic derivatives — The rule page lists this example and the others in the same family.
Derivative of ln(x) — Same family of differentiation, different function.
Derivative of log(x) — Same family of differentiation, different function.

Frequently asked questions

What is the derivative of ln(2x)?

x^(-1). ln(2x) looks like it should carry a 2, but the log property says ln(2x) = ln 2 + ln x. The derivative of the constant ln 2 is 0, so the derivative is 1/x, the same as the derivative of ln x. The chain rule agrees: the derivative of ln(u) is 1/u, times u' = 2, and 2/(2x) cancels to 1/x. The 2 is real and then it cancels.