Name: e from natural log derivatives
Uploaded: 2026-07-22T07:15:32.192Z
Duration: 1 min 29 s
Channel: JoVE

숫자 e 는 미적분학에서 기본적인 수학적 상수로, 특히 지수적 성장의 맥락에서 그렇습니다.그 역함수는 자연로그로, e를 보다 정확히 정의하는 데 도움을 준다.자연 로그 함수의 미분은 1을 입력 x로 나눈 값이다. x가 1일 때, 이 미분은 정확히 1이다. 이 관계는 e를 한계까지 정의하는 기초가 된다.미분은 미적분학의 1원리를 사용하여 표현할 수 있으며, 여기서 h는 x의 작은 변화를 나타냅니다. 1의 자연로그가 0이므로, 이 식은 단순화된다.양측 모두 계수를 계산하면 로그를 제거합니다. 작은 변화의 변수로 x를 사용하면 e의 첫 번째 극한이 정의됩니다. x가 0에 가까워질수록 이 식의 값은 상수 값 e에 가까워집니다.e를 정의하는 또 다른 방법은 분모가 커질수록 감소하는 분수로 입력값을 대체하는 것입니다. 이 비율이 감소하면 표현식은 e에서 안정됩니다.이 정의는 널리 사용되고 있습니다. 예를 들어, 연속 복리 이자에서 복리 빈도가 무한대에 가까워질수록 한계는 e가 되어 금액이 지수적으로 증가합니다.

극한으로 정의되는 자연상수 e

&#xC790;&#xC5F0;&#xC0C1;&#xC218; e&#xB294; &#xBBF8;&#xC801;&#xBD84;&#xD559;&#xC5D0;&#xC11C; &#xD575;&#xC2EC; &#xC0C1;&#xC218;&#xB85C;&#xC11C;, &#xD2B9;&#xD788; &#xC9C0;&#xC218; &#xC131;&#xC7A5; &#xB4F1; &#xC5F0;&#xC18D; &#xBCC0;&#xD654;&#xB97C; &#xC124;&#xBA85;&#xD558;&#xB294; &#xB370; &#xC911;&#xC2EC;&#xC801;&#xC778; &#xC5ED;&#xD560;&#xC744; &#xD569;&#xB2C8;&#xB2E4;. e&#xB294; &#xBC11;&#xC774; e&#xC778; &#xC9C0;&#xC218;&#xD568;&#xC218;&#xC758; &#xC5ED;&#xD568;&#xC218;&#xC778; &#xC790;&#xC5F0;&#xB85C;&#xADF8;&#xD568;&#xC218;&#xC640;&#xC758; &#xAD00;&#xACC4;&#xB97C; &#xD1B5;&#xD574; &#xAC00;&#xC7A5; &#xC790;&#xC5F0;&#xC2A4;&#xB7FD;&#xAC8C; &#xC815;&#xC758;&#xB429;&#xB2C8;&#xB2E4;. &#xC774;&#xB7EC;&#xD55C; &#xAD00;&#xACC4;&#xB97C; &#xD1B5;&#xD574; e&#xB294; &#xC784;&#xC758;&#xC758; &#xC218;&#xCE58; &#xC0C1;&#xC218;&#xAC00; &#xC544;&#xB2C8;&#xB77C;, &#xBBF8;&#xBD84;&#xC758; &#xAE30;&#xBCF8; &#xC6D0;&#xB9AC;&#xC5D0; &#xADFC;&#xAC70;&#xD558;&#xC5EC; &#xC815;&#xC758;&#xB420; &#xC218; &#xC788;&#xC2B5;&#xB2C8;&#xB2E4;.&#xC790;&#xC5F0;&#xB85C;&#xADF8; &#xD568;&#xC218; ln(x)&#xC758; &#xD575;&#xC2EC;&#xC801;&#xC778; &#xC18D;&#xC131;&#xC740; &#xB3C4;&#xD568;&#xC218;&#xAC00; &#xC785;&#xB825;&#xAC12;&#xC758; &#xC5ED;&#xC218;&#xC640; &#xAC19;&#xB2E4;&#xB294; &#xAC83;&#xC785;&#xB2C8;&#xB2E4;. &#xD2B9;&#xD788;, x&#xAC00; 1&#xC77C; &#xB54C; &#xC774; &#xB3C4;&#xD568;&#xC218;&#xB97C; &#xACC4;&#xC0B0;&#xD558;&#xBA74; &#xACB0;&#xACFC;&#xB294; &#xC815;&#xD655;&#xD788; 1&#xC785;&#xB2C8;&#xB2E4;. &#xC774; &#xC870;&#xAC74;&#xC740; &#xC790;&#xC5F0;&#xC9C0;&#xC218; &#xD568;&#xC218;&#xC758; &#xBC11;&#xC744; &#xACE0;&#xC720;&#xD558;&#xAC8C; &#xACB0;&#xC815;&#xD558;&#xBA70;, e&#xB97C; &#xC815;&#xC758;&#xD558;&#xB294; &#xB370; &#xC788;&#xC5B4; &#xC5C4;&#xBC00;&#xD55C; &#xAE30;&#xCD08;&#xB97C; &#xC81C;&#xACF5;&#xD569;&#xB2C8;&#xB2E4;.&#xBBF8;&#xBD84;&#xC758; &#xC81C;1&#xC6D0;&#xB9AC;&#xB97C; &#xC0AC;&#xC6A9;&#xD558;&#xBA74;, x = 1&#xC5D0;&#xC11C; ln(x)&#xC758; &#xB3C4;&#xD568;&#xC218;&#xB294; &#xC785;&#xB825;&#xAC12;&#xC758; &#xBBF8;&#xC18C;&#xD55C; &#xBCC0;&#xD654;&#xC5D0; &#xB300;&#xD55C; &#xADF9;&#xD55C;&#xC73C;&#xB85C; &#xD45C;&#xD604;&#xB429;&#xB2C8;&#xB2E4;. 1&#xC758; &#xC790;&#xC5F0;&#xB85C;&#xADF8; &#xAC12;&#xC774; 0&#xC774;&#xAE30; &#xB54C;&#xBB38;&#xC5D0;, &#xACB0;&#xACFC;&#xC801;&#xC73C;&#xB85C; &#xC5BB;&#xC5B4;&#xC9C0;&#xB294; &#xC2DD;&#xC740; &#xC0C1;&#xB2F9;&#xD788; &#xAC04;&#xB2E8;&#xD574;&#xC9D1;&#xB2C8;&#xB2E4;. &#xB2E8;&#xC21C;&#xD654;&#xB41C; &#xC2DD;&#xC758; &#xC591;&#xBCC0;&#xC5D0; &#xC9C0;&#xC218;&#xD568;&#xC218;&#xB97C; &#xCDE8;&#xD558;&#xBA74; &#xB85C;&#xADF8; &#xD56D;&#xC774; &#xC81C;&#xAC70;&#xB418;&#xC5B4;, &#xC77C;&#xC815;&#xD55C; &#xAC12;&#xC5D0; &#xC218;&#xB834;&#xD558;&#xB294; &#xADF9;&#xD55C; &#xD45C;&#xD604;&#xC744; &#xC5BB;&#xAC8C; &#xB429;&#xB2C8;&#xB2E4;. &#xC774; &#xACFC;&#xC815;&#xC5D0;&#xC11C; &#xB2E4;&#xC74C;&#xACFC; &#xAC19;&#xC740; &#xC815;&#xC758;&#xC2DD;&#xC774; &#xB3C4;&#xCD9C;&#xB429;&#xB2C8;&#xB2E4;.\begin{equation*} e = \lim_{x \to 0} (1+x)^{1/x} \end{equation*}&#xBCC0;&#xC218;&#xC758; &#xBCC0;&#xD654;&#xB7C9;&#xC774; &#xC784;&#xC758;&#xB85C; &#xC791;&#xC544;&#xC9D0;&#xC5D0; &#xB530;&#xB77C; &#xC774; &#xC2DD;&#xC740; &#xC0C1;&#xC218; e&#xB85C; &#xC218;&#xB834;&#xD569;&#xB2C8;&#xB2E4;.&#xB610; &#xB2E4;&#xB978;, &#xADF8;&#xB7EC;&#xB098; &#xB3D9;&#xCE58;&#xC778; &#xC815;&#xC758;&#xB294; &#xBBF8;&#xC18C;&#xD55C; &#xBCC0;&#xD654;&#xB7C9;&#xC744; &#xCEE4;&#xC9C0;&#xB294; &#xC815;&#xC218;&#xC758; &#xC5ED;&#xC218;&#xB85C; &#xB300;&#xCCB4;&#xD558;&#xC5EC;, &#xB110;&#xB9AC; &#xC0AC;&#xC6A9;&#xB418;&#xB294; &#xB2E4;&#xC74C;&#xACFC; &#xAC19;&#xC740; &#xD615;&#xD0DC;&#xB97C; &#xC81C;&#xACF5;&#xD569;&#xB2C8;&#xB2E4;.\begin{equation*} e = \lim_{n \to \infty} \liparens{1+\jfrac{1}{n}}^n \end{equation*}&#xC774;&#xC640; &#xAC19;&#xC740; &#xC815;&#xC758;&#xB294; &#xD2B9;&#xD788; &#xC751;&#xC6A9; &#xBD84;&#xC57C;&#xC5D0;&#xC11C; &#xC911;&#xC694;&#xD55C; &#xC758;&#xBBF8;&#xB97C; &#xAC00;&#xC9D1;&#xB2C8;&#xB2E4;. &#xC608;&#xB97C; &#xB4E4;&#xC5B4;, &#xC5F0;&#xC18D;&#xBCF5;&#xB9AC; &#xACC4;&#xC0B0;&#xC5D0;&#xC11C;&#xB294; &#xBCF5;&#xB9AC; &#xACC4;&#xC0B0;&#xC758; &#xBE48;&#xB3C4;&#xB97C; &#xC810;&#xC810; &#xC99D;&#xAC00;&#xC2DC;&#xD0AC;&#xC218;&#xB85D; &#xB204;&#xC801; &#xAE08;&#xC561;&#xC740; &#xBC11;&#xC774; e&#xC778; &#xC9C0;&#xC218;&#xD568;&#xC218;&#xB85C; &#xC218;&#xB834;&#xD569;&#xB2C8;&#xB2E4;. &#xC774;&#xCC98;&#xB7FC; &#xC0C1;&#xC218; e&#xB294; &#xC774;&#xB860;&#xC801;&#xC778; &#xBBF8;&#xC801;&#xBD84;&#xD559;&#xBFD0;&#xB9CC; &#xC544;&#xB2C8;&#xB77C;, &#xC2E4;&#xC81C; &#xC5F0;&#xC18D; &#xC131;&#xC7A5; &#xD604;&#xC0C1;&#xC744; &#xC124;&#xBA85;&#xD558;&#xB294; &#xBAA8;&#xB378;&#xC5D0;&#xC11C;&#xB3C4; &#xC790;&#xC5F0;&#xC2A4;&#xB7FD;&#xAC8C; &#xB4F1;&#xC7A5;&#xD569;&#xB2C8;&#xB2E4;.

&#xC790;&#xC5F0;&#xC0C1;&#xC218; e&#xB294; &#xBBF8;&#xC801;&#xBD84;&#xD559;&#xC5D0;&#xC11C; &#xD575;&#xC2EC; &#xC0C1;&#xC218;&#xB85C;&#xC11C;, ...

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Education

교육

JoVE Core

Calculus

미적분학

Unit

단위

Differentiation Rules

미분 규칙

KEY TERMS AND CONCEPTS

주요 용어 및 개념

The Number e as a Limit

수 e를 극한으로

e from natural log derivatives

The number e is a key constant in calculus, and it helps describe continuous change. It is especially important for exponential growth. Rather than treating e as an arbitrary number, the transcript shows how it can be defined through the natural logarithm and differentiation.

The natural logarithm, written as ln x, is the inverse of the exponential function with base e. Its derivative is 1 divided by its input. When that derivative is evaluated at x = 1, the result is exactly 1. This condition gives a precise way to identify the base of the natural exponential function.

Using the first principle of calculus, the derivative of ln x at x = 1 can be written as a limit with a very small change in x. Because ln 1 = 0, the expression simplifies. After exponentiating, the logarithm is removed, and the limit approaches a constant value. That constant is e.

The same idea can also be written with the small change replaced by 1 over a growing integer. This gives a common limit form for e. The result is useful in applied math, including continuously compounded interest, where faster and faster compounding makes the amount approach an exponential function with base e. In this way, e appears naturally in both calculus and real-world growth models.

e from Natural Log Derivatives

derivatives of simple functions

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limits on trigonometric functions

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the chain rule

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implicit differentiation

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implicit differentiation: problem solving

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the number e as a limit

The number e is a key constant in calculus, and it helps describe continuous change. It is especially important for exponential growth. Rather than treating e as an arbitrary number, the transcript shows how it can be defined through the natural logarithm and differentiation. The natural logarithm, written as ln x, is the inverse of the exponential function with base e. Its derivative is 1 divided by its input. When that derivative is evaluated at x = 1, the result is exactly 1. This condition...

derivatives of inverse trigonometric functions

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inverse hyperbolic functions and their derivatives

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Deriving the Slope of a Catenary Curve

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key terms and concepts

Learn how the constant e can be defined using the derivative of ln x and a limit. See why e matters for continuous growth and compounded interest.

e from Natural Log Derivatives | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

e from Natural Log Derivatives

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Transcript