What an Exponential Function is?
In general language, the exponential function is a sort of complex mathematical model that is to calculate the development or depletion of a population, a price that grows or falls exponentially.
This linking relationship is described by a function, which is commonly stated in a formula and defines how a particular component from the domain is connected to precisely one other component from the range.
Mathematical Definitions of Exponential Function: The fundamental form of an exponential function is f (x) = a
^{x}, where a is larger than unit zero, and just not equal to unit 1. The exponential function really shouldn’t be mistaken and misunderstood with polynomial functions. The label of the function is one method to tell the distinction between the two functions. Exponential functions get their name from the fact that the variable is contained inside the exponent of the functional operation.
History of Exponential Function
“Exactly how long would it take for something like a value of the currency to double if kept invested at a 40% rate of interest annualized return?”, this is the only question on the stone slab that is asked by many mathematicians.
When eminent mathematicians like Leonard Euler, and Jacob Bernoulli began to probe into the intricacies of exponential and logarithmic operations in the seventeenth and eighteenth centuries, exponential function studies became more intense. Their discoveries led to the development of contemporary calculus, which provided meaning to a slew of mathematical phenomena previously described exclusively by exponential functions. Exponential functions are intriguing to study and apply in everyday life.
Exponential Function Derivatives
Starting with a basic exponential curve, f (x) = (c)a
^{x}, with function y = f (x). One can see that variable, a represents a constant foundation increased towards the x power, and c represents a fixed value.
Exponential Functions: Range and Domain
An exponential function’s domain is the collection of all x-variables for something which can be estimated, whereas its range is indeed the collection of all y-variables.
It’s clear that an exponential function may be calculated for any value of x by looking at the plotted graphs of f(x), and g(x).
As a result, a function’s domain is the collection of all absolute values (real numbers). The equilibrium point of the curve, suppose y = d, and whether the plot is below or above y = d may be used to calculate the range of the function.
Exponential Function Derivative
Here are a few formulae for finding the derivative of an exponential distribution using differentiation algorithms.
- d/dx (a^{x}) = a^{x} · ln a
- d/dx (e^{x}) = e^{x}
Exponential Function Integration
Integration formulas for finding the integral of an exponential function are as follows:
- ∫ a^{x} dx = a^{x} / (ln a) + C
- ∫ e^{x} dx = e^{x} + C
Types of Exponential Functions
There are two types of exponential functions in general, these are as follows:
In general language, in exponential growth, a quantity rises slowly at first before rapidly increasing.
In general language, in exponential decay, a quantity drops slowly at first before rapidly decreasing.
Conclusion
Conclusively, it is clear that with exponential functions we can measure the growth and decay in some aspects, for instance, in population, currency valuation, and in so many elements. With the help of depreciation and integration of exponential functions, we could even estimate the future estimated growth of many variables and this is used everywhere, majorly to estimate the growth of the overall economy.