e

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