e

(noun)

the base of the natural logarithm, $2.718281828459045\dots$

Related Terms

  • exponential
  • logarithm
  • tangent
  • differentiation

Examples of e in the following topics:

  • The Natural Exponential Function: Differentiation and Integration

    • First, we determine the derivative of $e^{x}$ using the definition of the derivative:
    • Since $e^{x}$ does not depend on $h$, it is constant as $h$ goes to $0$.
    • Therefore $\ln(e^x) = x$ and $e^{\ln x} = x$.
    • Let's consider the example of $\int e^{x}dx$.
    • Since $e^{x} = (e^{x})'$ we can integrate both sides to get:
  • Hyperbolic Functions

    • $\sinh$ and $\cosh$ are basic hyperbolic functions; $\sinh$ is defined as the following: $\sinh (x) = \frac{e^x - e^{-x}}{2}$.
    • $\tanh (x) = \dfrac{\sinh(x)}{\cosh (x)} = \dfrac{1 - e^{-2x}}{1 + e^{-2x}}$
    • $\coth (x) = \dfrac{\cosh (x)}{\sinh (x)} = \dfrac{1 + e^{-2x}}{1 - e^{-2x}}$
  • Differentiation Rules

    • $\displaystyle{f'(x) = 4x^{(4-1)} + \frac{d(x^2)}{dx}e^{x^2} - \frac {d(ln\:x)}{dx}e^x - ln\:x\frac{d(e^x)}{dx} + 0 \\ \qquad = 4x^3 + 2xe^{x^2} - \frac {1}{x}e^x - ln(x)e^x}$
    • The known derivatives of the elementary functions $x^2$, $x^4$, $\ln(x)$, and $e^x$, as well as that of the constant 7, were also used.
  • Exponential and Logarithmic Functions

    • Exponential function is the function $e^x$ the number (approximately 2.718281828) such that the function $e^x$ is its own derivative .
    • The exponential function $e^x$ can be characterized in a variety of equivalent ways.
    • In particular it may be defined by the following power series: $\displaystyle e^x = \sum_{n = 0}^{\infty} {x^n \over n!
    • From this definition, you can check that $e^x$ is its own derivative: $\displaystyle \frac{d}{dx} e^x = e^x$.
    • For $f(x)=e^x$, $g(x)=\log_e(x)$ is the inverse function of $f(x)$ and vice versa.
  • Conic Sections in Polar Coordinates

    • where e is the eccentricity and l is half the latus rectum.
    • As in the figure, for $e = 0$, we have a circle, for $0 < e < 1$ we obtain an ellipse, for $e = 1$ a parabola, and for $e > 1$ a hyperbola.
  • Derivatives of Exponential Functions

  • Derivatives of Logarithmic Functions

    • First, we will derive the equation for a specific case (the natural log, where the base is $e$), and then we will work to generalize it for any logarithm.
    • Next, we will raise both sides to the power of $e$ in an attempt to remove the logarithm from the right hand side:
    • Substituting back our original equation of $x = e^{y}$, we find that
  • Logistic Equations and Population Grown

    • Choosing the constant of integration $e^c = 1$ gives the other well-known form of the definition of the logistic curve:
  • The Divergence Theorem

    • The first equation of the Maxwell's equations is often written as $\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}$ in a differential form, where $\rho$ is the electric density.
    • Substituting $E$ for $F$ in the relationship of the divergence theorem, the left hand side (LHS) becomes:
  • Stokes' Theorem

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