Mathematical tables: exponential function formulas
Table show common formulas related to exponential function.

Beta version

BETA TEST VERSION OF THIS ITEM
This online calculator is currently under heavy development. It may or it may NOT work correctly.
You CAN try to use it. You CAN even get the proper results.
However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.
Feel free to send any ideas and comments !
⌛ Loading...

Various forms of function formulas

NameFormulaLegend
Exponential function in general formShow sourcey=ax y={ a}^{ x}
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa - base of the exponential function.
Exponential function with base e (often written as exp(x))Show sourceexp(x)=ex exp(x)={ e}^{ x}
  • exp(x)exp(x) - value of exponent function,
  • xx - function argument (called also independent value),
  • ee - number e (mathematical constant, base of natural logarithm).
Homographic function in general formShow sourcey=a x+bc x+d y=\frac{ a~ x+ b}{ c~ x+ d}
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • a, b, c, d - homographic function coefficients (parameters defining concrete homographic function, c ≠ 0).
Function b/xShow sourcey=bx y=\frac{ b}{ x}
  • yy - value of b/x function (the value of f(x)=b/x function for given x, parameters a,d are zero, parameter c is 1),
  • xx - function argument (called also independent value),
  • b - coefficient b.
Linear function in slope-intercept formShow sourcey=a x+b y= a~ x+ b
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa, bb - linear function coefficients (slope and free parameter).
Linear function in point-slope formShow sourcey=a(xx0)+y0 y=\mathrm{a}\left( x- x_0\right)+ y_0
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa - slope (number that describes the direction and the steepness of the line, sometimes is called gradient),
  • x0x_0, y0y_0 - point coordinates.
Linear function in constant-slope formShow sourceyy0xx0=y1y0x1x0\frac{y - y_0}{x - x_0} = \frac{y_1 - y_0}{x_1 - x_0}
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • x0x_0, y0y_0 - coordinates of the first point,
  • x1x_1, y1y_1 - coordinates of the second point.
Zero of the linear function from constant-slope formShow sourcex=y0(x1x0)y1y0+x0 x=\frac{ y_0\cdot\left( x_1- x_0\right)}{ y_1- y_0}+ x_0
  • xx - zero of the function (argument for which the function has a value of zero, its a solution of f(x) = 0 equation),
  • x0x_0, y0y_0 - coordinates of the first point,
  • x1x_1, y1y_1 - coordinates of the second point.
Quadratic function in standard formShow sourcey=a x2+b x+c y= a~{ x}^{2}+ b~ x+ c
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa, bb, cc - quadratic function coefficients (numbers just before x2, x and free parameter).
Quadratic function in factored formShow sourcey=a(xx1) (xx2) y=\mathrm{a}\left( x- x_1\right)~\left( x- x_2\right)
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa - coefficient before power of two (number just before x2),
  • x1x_1, x2x_2 - function zero points (arguments, for which function has value of zero, solutions of the f(x)=0 equation).
Quadratic function in vertex formShow sourcey=a(xh)2+k y={\mathrm{a}\left( x- h\right)}^{2}+ k
  • yy - function value (the function value at single point x, often marked as f(x)),
  • xx - function argument (called also independent value),
  • aa - coefficient before power of two (number just before x2),
  • hh, kk - coordinates of the parabola vertex (at this point function reaches its local extremum).

Function properties

NameFormulaLegend
Product of two exponential functionsShow sourceaxay=ax+ya ^ {x} \cdot a ^ {y} = a ^ {x + y}
  • xx - the first argument,
  • yy - the second argument,
  • aa - base of the exponential function.
Quotient of two exponential functionsShow sourceaxay=axy\frac{a ^ {x}}{a ^ {y}} = a ^ {x - y}
  • xx - the first argument,
  • yy - the second argument,
  • aa - base of the exponential function.

Derivatives and integrals

NameFormulaLegend
Derivative of general exponential functionShow sourceddxax=axln(a)\frac{d}{d x} a^{x} = a^{x} \ln(a)
  • xx - function argument (called also independent value),
  • aa - base of the exponential function.
Derivative of exponential function with e base, derivative of exp(x) functionShow sourceddxex=exln(e)=ex\frac{d}{d x} e^{x} = e^{x} \ln(e) = e^{x}
Indefinite integral of general exponential functionShow sourceaxdx=axln(a)\int a^{x} dx = \frac{a^{x}}{\ln(a)}
  • xx - function argument (called also independent value),
  • aa - base of the exponential function.
Indefinite integral of exponential function with e base, integral of exp(x) functionShow sourceexdx=ex\int e^{x} dx = e^{x}

Some facts

  • Exponential function is a function that can be presented in the form:
    y=ax y={ a}^{ x}
    where:
    • yy - function value (the function value at single point x, often marked as f(x)),
    • xx - function argument (called also independent value),
    • aa - base of the exponential function.
  • A special case is the exponential function of the base e (→ see the number e):
    exp(x)=ex exp(x)={ e}^{ x}
  • The exponential function with the base e is often denoted as exp (x), which we read as exponent of x.
  • The inverse function for the exponential one is logarithmic function. In particular for the function exp(x) (the base is number e) the inverse function is natural logarithm.
  • The exponential function has no zero poins. Its all values ​​are located above the OX axis (all function values are positive).
  • Depending on the base a, we can distinguish three scenarios:
    • base is less than one (a <1) - function is decreasing,
    • base is greater than one (a > 1) - the function is inreasing,
    • base is equal to zero (a = 0) - the function is reduced to constant function.

Tags and links to this website

What tags this calculator has

Permalink

This is permalink. Permalink is the link containing your input data. Just copy it and share your work with friends:

Links to external sites (leaving Calculla?)

JavaScript failed !
So this is static version of this website.
This website works a lot better in JavaScript enabled browser.
Please enable JavaScript.