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

The number e is a fundamental mathematical constant in calculus, particularly in the context of exponential growth.Its inverse function is the natural logarithm, which helps define e more precisely.The derivative of the natural log function is one divided by its input x. At x equals one, this derivative is exactly one. This relationship forms the basis for defining e through limits.The derivative can be expressed using its limit definition, where h represents a small change in x. Since the natural logarithm of one is zero, the expression simplifies.Exponentiating both sides removes the logarithm. Using x as the variable for the small change gives the first limit definition of e. As x approaches zero, the value of this expression approaches a constant value, e.Another way to define e is to replace the input with a fraction that decreases as its denominator increases. As this fraction decreases, the expression stabilizes at e.This definition is widely used. For example, in continuously compounded interest, as the compounding frequency approaches infinity, the limit becomes e, leading to exponential growth in the amount.

the number e as a limit

The number e is a fundamental constant in calculus, playing a central role in describing continuous change, particularly exponential growth. It is most naturally defined through its relationship with the natural logarithm, which is the inverse of the exponential function with base e. This relationship allows e to be characterized using basic principles of differentiation rather than as an arbitrary numerical constant.A key property of the natural logarithm function, ln x, is that its derivative is equal to the reciprocal of its input. In particular, when this derivative is evaluated at x equals to 1, the result is exactly one. This condition uniquely determines the base of the natural exponential function and provides a rigorous foundation for defining e.Using the first principle of calculus, the derivative of ln &#x2061;x at x equals 1 is expressed as a limit involving a small change in the input. Because the natural logarithm of one is zero, the resulting expression simplifies significantly. By exponentiating the simplified form, the logarithm is eliminated, yielding a limit that approaches a constant value. This leads to the defining expression<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:25%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19804/PT_19804_1_editor_19804.svg" alt="PT_19804_1.svg" /></figure>As the change in the variable becomes arbitrarily small, the expression converges to the constant e.An alternative but equivalent definition replaces the small increment with the reciprocal of a growing integer, producing the widely used form<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:36.33%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19804/PT_19804_2_editor_19804.svg" alt="PT_19804_2.svg" /></figure>This formulation is particularly important in applied contexts. In continuously compounded interest, for example, increasing the compounding frequency causes the accumulated amount to approach an exponential function with base e. In this way, the constant e emerges naturally from both theoretical calculus and real-world models of continuous growth.

The Number e as a Limit

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Calculus

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Differentiation Rules

KEY TERMS AND CONCEPTS

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

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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...

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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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