TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t

Statement

For $x>0$ , $\log$ is differentiable and $\log'(x)=\frac1x,\qquad \log x=\int_1^x\frac{dt}{t}.$

Facts & Assumptions

Given: A positive real $x$ .

[L1]

If a differentiable injective function has nonzero derivative, its inverse is differentiable at an image point and its derivative is the reciprocal of the original derivative (Derivative of an inverse: if $f$ is continuous and injective on a nondegenerate interval $I$ and differentiable at $c \in I$ with $f'(c) \ne 0$ , then the inverse $g$ is differentiable at $f(c)$ with $g'(f(c)) = 1/f'(c)$ ; and if $f'(c) = 0$ then $g$ is not differentiable at $f(c)$ ).

[L2]

$\exp'(u)=\exp(u)$ for every real $u$ (The exponential function is smooth and $(\exp)'=\exp$ ).

[L3]

The reciprocal of a nonvanishing continuous real function is continuous, and the first FTC differentiates $x\mapsto\int_1^x f(t)\,dt$ when $f$ is continuous (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, The first fundamental theorem: if $f$ is integrable on $[a,b]$ and continuous at $c$ , then $F'(c) = f(c)$ ; in particular a continuous $f$ has $F$ as a primitive, The integral with oriented limits: $\int_a^a f := 0$ and $\int_b^a f := -\int_a^b f$ ).

[L4]

A differentiable real function with zero derivative on an interval is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant).

Proof

technique · direct

1.1

At $u=\log x$ , $\exp'(u)=\exp(u)=x\ne0$ ; hence the inverse rule gives $\log'(x)=1/\exp(\log x)=1/x$ .

L1L2

1.2

The function $t\mapsto1/t$ is continuous on $(0,\infty)$ , so $I(x):=\int_1^xdt/t$ satisfies $I'(x)=1/x$ and $I(1)=0$ .

2.1

The difference $x\mapsto\log x-I(x)$ has derivative zero on $(0,\infty)$ , and is zero at $1$ , so it is identically zero.

step 1.1step 1.2L4∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 124 results over 25 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)