Derivative of log(x)

Answer

\frac{1}{x}

Derivative of ln(x)

Rule Used: Logarithmic Rule
Steps: Recognize the natural logarithm $\log(x)$ , written $\ln(x)$ on this site. Its derivative is the reciprocal because ln is the inverse of e^x and the derivative of e^x is itself.
Apply the rule: $\frac{d}{dx} \log(x)$ = 1/x.
Final answer: $x^{-1}$

The derivative of log(x), taken as the natural logarithm, is 1/x, defined only for x > 0.

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

Derivative LogarithmicIntermediate

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

Derivative

log(x) → 1/xengine: 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

On this calculator log(x) means the natural logarithm, the same function as ln(x). The derivative is 1/x. Base-10 log is written log10(x), and that one does carry a factor 1/ln(10). Mixing the two notations is the point of this page: the symbol log is not automatically base 10 here, because the math engine treats log as ln.

Apply the logarithm rule to log(x). The form to remember, already simplified, is 1/x.

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

Worked derivative

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

Common mistakes

Answering 1/(x·ln 10) instead of 1/x

1/(x ln 10) is the derivative of log10(x), not of the natural log that this page computes. If you wanted base 10, the input has to be log10(x).

Confusing the base of log(x)

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

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 ln(2x) — Same family of differentiation, different function.

Frequently asked questions

What is the derivative of log(x)?

1/x. On this calculator log(x) means the natural logarithm, the same function as ln(x). The derivative is 1/x. Base-10 log is written log10(x), and that one does carry a factor 1/ln(10). Mixing the two notations is the point of this page: the symbol log is not automatically base 10 here, because the math engine treats log as ln.