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 exponential is the unique solution of with
Statement
If is differentiable, , and , then .
Facts & Assumptions
Given: A differentiable solution of the initial-value problem.
, , and the series definition gives (The exponential function is smooth and , The exponential is positive and satisfies , The real exponential function and the number by a power series).
Differentiability implies continuity, and a continuous function with zero derivative on an interval is constant (A function differentiable at is continuous at , A function continuous on an interval whose derivative vanishes at every interior point of is constant on ; consequently two such functions with the same derivative differ by a constant).
Proof
Define . By [L1] and [L2], .
The differentiable function is continuous, so [L3] makes it constant; .
Thus , and multiplying by gives .
Depends on
- The real exponential function and the number $e$ by a power series
- The exponential function is smooth and $(\exp)'=\exp$
- The exponential is positive and satisfies $\exp(-x)=1/\exp(x)$
- Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$, $(\alpha f)'(c) = \alpha f'(c)$, $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$, and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$
- The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f \circ g$ is differentiable at $c$ with $(f \circ g)'(c) = f'(g(c))\,g'(c)$
- A function differentiable at $c$ is continuous at $c$
- 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
Used by
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 97 results over 22 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
- MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)
- J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
- J. K. Hunter, An Introduction to Real Analysis, Chapter 10 (standard reference, not scraped)
- University of Pennsylvania MATH 3600, Section 34 (standard reference, not scraped)