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

Liczba e jest fundamentalną stałą matematyczną w rachunku różniczkowym, szczególnie w kontekście wzrostu wykładniczego.Jego odwrotną funkcją jest logarytm naturalny, który pomaga precyzyjniej zdefiniować e.Pochodna naturalnej funkcji logarytmu to taka jak podzielona przez jej wejście x. Przy x równa się jeden, ta pochodna jest dokładnie jeden. Ta relacja stanowi podstawę do definiowania e przez granice.Pochodną można wyrazić za pomocą pierwszej zasady rachunku różniczkowego, gdzie h oznacza niewielką zmianę x. Ponieważ naturalny logarytm jeden jest równy zeru, wyrażenie upraszcza.Potęgowanie obu stron usuwa logarytm. Użycie x jako zmiennej dla małej zmiany daje pierwszą definicję granicy e. Gdy x zbliża się do zera, wartość tego wyrażenia zbliża się do stałej wartości, e.Innym sposobem zdefiniowania e jest zastąpienie wejścia ułamkiem, który maleje wraz ze wzrostem mianownika. W miarę maleka tego ułamku wyrażenie stabilizuje się w e.Ta definicja jest szeroko stosowana. Na przykład w ciągłym odsetku składanym, gdy częstotliwość kumulacji zbliża się do nieskończoności, limit staje się e, co prowadzi do wykładniczego wzrostu kwoty.

liczba e jako granica

Liczba e jest podstawow&#x105; sta&#x142;&#x105; w rachunku r&#xF3;&#x17C;niczkowym, odgrywaj&#x105;c&#x105; kluczow&#x105; rol&#x119; w opisie ci&#x105;g&#x142;ej zmiany, zw&#x142;aszcza wzrostu wyk&#x142;adniczego. Najnaturalniej definiuje si&#x119; j&#x105; przez zwi&#x105;zek z logarytmem naturalnym, kt&#xF3;ry jest odwrotno&#x15B;ci&#x105; funkcji wyk&#x142;adniczej o podstawie e. Ten zwi&#x105;zek pozwala na scharakteryzowanie e za pomoc&#x105; podstawowych zasad r&#xF3;&#x17C;niczkowania, a nie jako dowolnej sta&#x142;ej liczbowej.Kluczow&#x105; w&#x142;a&#x15B;ciwo&#x15B;ci&#x105; funkcji logarytmu naturalnego ln x jest to, &#x17C;e jej pochodna jest r&#xF3;wna odwrotno&#x15B;ci jej argumentu. W szczeg&#xF3;lno&#x15B;ci, gdy ta pochodna jest obliczana w punkcie x = 1, wynik jest r&#xF3;wny dok&#x142;adnie 1. Ten warunek jednoznacznie okre&#x15B;la podstaw&#x119; naturalnej funkcji wyk&#x142;adniczej i stanowi &#x15B;cis&#x142;&#x105; podstaw&#x119; do zdefiniowania e.Korzystaj&#x105;c z definicji granicznej pochodnej, pochodna ln x dla x = 1 jest wyra&#x17C;ona jako granica obejmuj&#x105;ca niewielk&#x105; zmian&#x119; warto&#x15B;ci wej&#x15B;ciowej. Poniewa&#x17C; logarytm naturalny z liczby jeden jest r&#xF3;wny zeru, otrzymane wyra&#x17C;enie ulega znacznemu uproszczeniu. Przez pot&#x119;gowanie uproszczonej formy logarytm zostaje wyeliminowany, co daje granic&#x119; d&#x105;&#x17C;&#x105;c&#x105; do warto&#x15B;ci sta&#x142;ej. Prowadzi to do wyra&#x17C;enia definiuj&#x105;cego.\begin{equation*} e = \lim_{x \to 0} (1+x)^{1/x} \end{equation*}Gdy zmiana zmiennej staje si&#x119; dowolnie ma&#x142;a, wyra&#x17C;enie zbiega do sta&#x142;ej r&#xF3;wnej e.Alternatywna, ale r&#xF3;wnowa&#x17C;na definicja zast&#x119;puje ma&#x142;y przyrost odwrotno&#x15B;ci&#x105; rosn&#x105;cej liczby ca&#x142;kowitej, prowadz&#x105;c do powszechnie stosowanej postaci:\begin{equation*} e = \lim_{n \to \infty} \liparens{1+\jfrac{1}{n}}^n \end{equation*}To sformu&#x142;owanie jest szczeg&#xF3;lnie wa&#x17C;ne w zastosowaniach praktycznych. Na przyk&#x142;ad, w przypadku odsetek kapitalizowanych w spos&#xF3;b ci&#x105;g&#x142;y, zwi&#x119;kszenie cz&#x119;stotliwo&#x15B;ci kapitalizacji powoduje, &#x17C;e skumulowana kwota zbli&#x17C;a si&#x119; do funkcji wyk&#x142;adniczej o podstawie e. W ten spos&#xF3;b sta&#x142;a e pojawia si&#x119; naturalnie zar&#xF3;wno w rachunku teoretycznym, jak i w rzeczywistych modelach wzrostu ci&#x105;g&#x142;ego.

Liczba e jako granica

Liczba e jest podstawow&#x105; sta&#x142;&#x105; w rachunku r&#xF3;&#x17C;niczkowym, odgrywaj&#x105;c&#x105; kluczow&#x105; rol&#x119; w opisie ...

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Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Differentiation Rules

Zasady różniczkowania

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

The Number e as a Limit

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

Slope Rules for Simple Functions

Derivatives show how fast a function changes, and they also describe slope. A derivative can tell the slope of a straight line or the slope of a tangent line to a curve at one point. In this lesson, the idea is tied to a roller coaster track, where the derivative of the horizontal position tells how steep the ride is at each spot. A flat roller coaster section can be modeled with a constant function, f(x) = c, where c is a fixed height. Its derivative is f'(x) = 0, so the slope is zero and...

the product rule

Derivative of a Changing Rectangle

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the quotient rule

Differentiating Ratios with the Quotient Rule

The quotient rule is a key calculus method for differentiating ratios of two differentiable functions. It is used when a function is written as one function divided by another, and the denominator is not zero. The rule gives a direct way to find the derivative of that ratio. The quotient rule is especially useful for rational functions, trigonometric ratios, and exponential functions. For example, it can be applied to a function in ratio form to find its derivative step by step. This makes it...

derivatives of the trigonometric functions

derivatives-of-the-trigonometric-functions

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limits on trigonometric functions

Trigonometric Limits for Derivatives

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the chain rule

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the chain rule: problem solving

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implicit differentiation

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second derivatives of implicit functions

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implicit differentiation: problem solving

Tangent Lines with Implicit Differentiation

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derivatives of logarithmic functions

Natural Log Derivatives and Real-World Uses

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logarithmic differentiation

Differentiating Tire Stress with Logs

Logarithmic differentiation is used to find how tire stress changes with strain. In this setting, a car’s weight and driving forces act as an external load on the rubber. The rubber resists that load through internal forces called stress. The deformation that results is called strain. The stress-strain relationship shows how the tire changes shape under load. For rubber, this relationship is nonlinear, so it does not behave like a linear elastic material. The model used here includes material...

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

derivatives-of-inverse-trigonometric-functions

Inverse Trig Derivatives in Tracking

Inverse trigonometric derivatives can describe how a tracking angle changes as an aircraft moves closer to an observer. In this situation, a ship measures the slant distance to the aircraft and the angle of elevation. These measurements let the observer find the horizontal and vertical parts of the distance using trigonometric relationships. The angle of elevation is treated as the dependent variable, and the ratio of altitude to slant distance is treated as the independent variable. That...

application of rates of change

Velocity and Acceleration in Motion Graphs

Velocity and acceleration describe how a car moves along a highway. The car’s position changes over time, so it can be written as a function of time. Rates of change help measure motion in a clear mathematical way. Velocity tells how position changes with time. Average velocity is found by dividing the change in position by the change in time. It gives the overall speed of the car during a time interval, but it does not show changes inside that interval. To find the velocity at one exact...

exponential growth

Bacterial Doubling in Exponential Growth

Bacterial populations can grow by doubling when conditions are favorable. When nutrients and temperature support growth, cells divide by binary fission. In binary fission, one cell splits into two identical daughter cells. This makes the population rise quickly. During exponential growth, the growth rate depends on how many cells are already present. As the population gets larger, more new cells are formed in each generation. The result is a steep increase in total cell count over time. The...

related rates

Balloon Inflation and Changing Rates

Balloon inflation is a clear example of related rates, where one changing quantity affects another through a single relationship. Related rates study how linked quantities change with respect to time. As air is added, the balloon’s volume rises, and the radius must also increase. For a hot air balloon, the inflated envelope is often modeled as a perfect sphere. This simplification makes the math easier to work with. In that model, the balloon’s volume depends directly on its radius. When air...

linearization and approximation

Tangent Line Estimates Near a Point

Linearization uses a tangent line to estimate a nonlinear function near a chosen reference point. It replaces a difficult function with a simpler linear model when the input changes only a little. This makes it useful for small deviations from a known value. A square root example shows how the method works. The value at 4 is known exactly, so 4 is a good reference point. The function value and the rate of change are both easy to find there. Estimating the value at a nearby input, such as 4.1,...

application of linearization and approximation

application-of-linearization-and-approximation

Estimating Altitude Changes with Linearization

Linearization can help a drone estimate small altitude changes from air pressure readings. Atmospheric pressure drops as altitude rises, and this relationship is often modeled with an exponential function. The model is accurate, but the math needed to convert pressure into altitude can be too complex for repeated calculations during flight. To make the calculation easier, the pressure-altitude relationship is approximated locally with a straight line. This line is built near a chosen reference...

hyperbolic functions

Modeling Hanging Cables with Hyperbolic Curves

Hyperbolic functions help describe the shape of a hanging cable. When a flexible cable is suspended between two points at the same height, it naturally forms a catenary. That curve comes from the balance between the cable’s weight and the tension along its length. It shows a state of mechanical equilibrium. The true shape of this hanging cable is not captured well by a simple approximation. Instead, it is described with hyperbolic functions. These functions are related to exponential functions.

inverse hyperbolic functions and their derivatives

inverse-hyperbolic-functions-and-their-derivatives

Deriving the Slope of a Catenary Curve

A catenary curve describes the shape of a suspension bridge cable hanging under its own weight. The curve is modeled with the hyperbolic cosine function, which helps show how gravity and tension act along the cable. When the vertical position on the cable is known, the matching horizontal position can be found with the inverse hyperbolic cosine function. Inverse hyperbolic functions are a family of functions that includes inverse hyperbolic sine, cosine, and tangent. Their corresponding...

hyperbolic and inverse hyperbolic functions: problem solving

hyperbolic-and-inverse-hyperbolic-functions-problem-solving

Tangent Lines on a Hyperbolic Arch

Hyperbolic cosine functions can model an arched gate with a smooth, symmetric shape. When the arch is centered at the origin, its highest point is at the center. That symmetry means any lower height appears at two horizontal points on opposite sides of the centerline. The arch is checked at a height of five meters by adjusting the function to match that level. Solving for the horizontal positions shows that there are exactly two points where the gate reaches five meters. These points are equal...

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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