Table show common formulas related to exponential function.

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Various forms of function formulas#

Name	Formula	Legend
Exponential function in general form	Show source $y=a^{x}$	$y$ - function value (the function value at single point x, often marked as f(x)), $x$ - function argument (called also independent value), $a$ - base of the exponential function.
Exponential function with base e (often written as exp(x))	Show source $exp(x)=e^{x}$	$exp(x)$ - value of exponent function, $x$ - function argument (called also independent value), $e$ - number e (mathematical constant, base of natural logarithm).
Homographic function in general form	Show source $y=\frac{a \cdot x+b}{c \cdot x+d}$	$y$ - function value (the function value at single point x, often marked as f(x)), $x$ - function argument (called also independent value), a, b, c, d - homographic function coefficients (parameters defining concrete homographic function, c ≠ 0).
Function b/x	Show source $y=\frac{b}{x}$	$y$ - 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), $x$ - function argument (called also independent value), b - coefficient b.
Linear function in slope-intercept form	Show source $y=a \cdot x+b$	$y$ - function value (the function value at single point x, often marked as f(x)), $x$ - function argument (called also independent value), $a$ , $b$ - linear function coefficients (slope and free parameter).
Linear function in point-slope form	Show source $y=a\left(x-x_0\right)+y_0$	$y$ - function value (the function value at single point x, often marked as f(x)), $x$ - function argument (called also independent value), $a$ - slope (number that describes the direction and the steepness of the line, sometimes is called gradient), $x_0$ , $y_0$ - point coordinates.
Linear function in constant-slope form	Show source $\frac{y - y_0}{x - x_0} = \frac{y_1 - y_0}{x_1 - x_0}$	$y$ - function value (the function value at single point x, often marked as f(x)), $x$ - function argument (called also independent value), $x_0$ , $y_0$ - coordinates of the first point, $x_1$ , $y_1$ - coordinates of the second point.
Zero of the linear function from constant-slope form	Show source $x=\frac{y_0 \cdot \left(x_1-x_0\right)}{y_1-y_0}+x_0$	$x$ - zero of the function (argument for which the function has a value of zero, its a solution of f(x) = 0 equation), $x_0$ , $y_0$ - coordinates of the first point, $x_1$ , $y_1$ - coordinates of the second point.
Quadratic function in standard form	Show source $y=a \cdot x^{2}+b \cdot x+c$	$y$ - function value (the function value at single point x, often marked as f(x)), $x$ - function argument (called also independent value), $a$ , $b$ , $c$ - quadratic function coefficients (numbers just before x², x and free parameter).
Quadratic function in factored form	Show source $y=a\left(x-x_1\right) \cdot \left(x-x_2\right)$	$y$ - function value (the function value at single point x, often marked as f(x)), $x$ - function argument (called also independent value), $a$ - coefficient before power of two (number just before x²), $x_1$ , $x_2$ - function zero points (arguments, for which function has value of zero, solutions of the f(x)=0 equation).
Quadratic function in vertex form	Show source $y=a\left(x-h\right)^{2}+k$	$y$ - function value (the function value at single point x, often marked as f(x)), $x$ - function argument (called also independent value), $a$ - coefficient before power of two (number just before x²), $h$ , $k$ - coordinates of the parabola vertex (at this point function reaches its local extremum).

Function properties#

Name	Formula	Legend
Product of two exponential functions	Show source $a ^ {x} \cdot a ^ {y} = a ^ {x + y}$	$x$ - the first argument, $y$ - the second argument, $a$ - base of the exponential function.
Quotient of two exponential functions	Show source $\frac{a ^ {x}}{a ^ {y}} = a ^ {x - y}$	$x$ - the first argument, $y$ - the second argument, $a$ - base of the exponential function.

Derivatives and integrals#

Name	Formula	Legend
Derivative of general exponential function	Show source $\frac{d}{d x} a^{x} = a^{x} \ln(a)$	$x$ - function argument (called also independent value), $a$ - base of the exponential function.
Derivative of exponential function with e base, derivative of exp(x) function	Show source $\frac{d}{d x} e^{x} = e^{x} \ln(e) = e^{x}$	$x$ - function argument (called also independent value), $e$ - number e (mathematical constant, base of natural logarithm).
Indefinite integral of general exponential function	Show source $\int a^{x} dx = \frac{a^{x}}{\ln(a)}$	$x$ - function argument (called also independent value), $a$ - base of the exponential function.
Indefinite integral of exponential function with e base, integral of exp(x) function	Show source $\int e^{x} dx = e^{x}$	$x$ - function argument (called also independent value), $e$ - number e (mathematical constant, base of natural logarithm).

Some facts#

Exponential function is a function that can be presented in the form:

$y=a^{x}$
where:
- $y$ - function value (the function value at single point x, often marked as f(x)),
- $x$ - function argument (called also independent value),
- $a$ - base of the exponential function.
A special case is the exponential function of the base e (→ see the number e):

$exp(x)=e^{x}$
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.

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