what is the purpose of the value e?
In: Mathematics
In algebra, you learn about functions and you’re mainly concerned with what the output is (y) for any given input (x).
In calculus, you are now concerned with the rate of change of the output (dy/dx) for any given input (x). In calculus, dy/dx is called “the derivative of y with respect to x”. Mathematically speaking, it is the slope of the function’s tangent line for any given input (x).
So what if we could make a function f(x) whose rate of change was the same as its output for all inputs? Well experimentally we can find that this is true of some specific number raised to the x power. That number is e (2.71…..).
For example: consider again the function y=e^x. let’s plug in 2 for our input x. This gives y = 7.39. What makes our function unique is that our output (y) is the same as our rate of change (dy/dx). So y= dy/dx = 7.39 This pattern will continue for all possible inputs.
So to recap, what we have created is a function whose rate of change increases exactly as its output does. In math terms, e^x is a function that is its own derivative. As the other commenters have pointed out, that has extremely important applications in compounding interest and estimating all sorts of cool stuff.
Calculus sounds much more intimidating than it is, and if you’d like a very very well explained and animated version of what calculus actually accomplishes, check out [this video ](https://youtu.be/WUvTyaaNkzM). Good question OP.
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