Exponential function

Based on Wikipedia: Exponential function

The Function That Equals Its Own Derivative

Here's a puzzle that sounds impossible: find a function whose rate of change at every point is exactly equal to its value at that point. If the function outputs 7 somewhere, its slope there must be exactly 7. If it outputs 1000, the slope must be 1000. Whatever the function spits out, that's how fast it's growing.

It turns out there's exactly one function that does this. Just one. And it's arguably the most important function in all of mathematics.

We call it the exponential function, written as e to the power of x, or sometimes exp of x. That letter e represents a specific number: approximately 2.718281828. It's not a round number, not a simple fraction, but an irrational constant that shows up everywhere in nature, finance, physics, and pure mathematics. The exponential function takes any number you give it and raises e to that power.

But what makes this function truly special isn't the number e itself. It's that remarkable property I mentioned: the exponential function is its own derivative. Take its derivative, and you get the same function back. Take the derivative again, same thing. Forever.

Why This Property Matters

Think about what it means for a function to equal its own rate of change. If you're growing at a rate proportional to your current size, you're experiencing exponential growth. Populations do this. Compound interest does this. Radioactive decay works the same way, just in reverse.

When a biologist says a bacterial colony is growing exponentially, they're not using the word loosely. They're saying the number of new bacteria appearing each hour depends on how many bacteria are already there. More bacteria means more reproduction means even more bacteria. The rate of change equals the current value. That's exactly what the exponential function describes.

This is why the function shows up in so many different fields. Any process where the speed of change is proportional to the current amount follows exponential behavior. Cooling coffee. Charging capacitors. The spread of rumors through a population. Nuclear chain reactions. The mathematics is the same underneath.

The Number e

So where does this mysterious number 2.718... come from?

Imagine you put one dollar in a bank that offers 100% annual interest. After one year, you'd have two dollars. Simple enough. But what if the bank compounds monthly? They'd give you one-twelfth of 100% each month, which is about 8.33% per month. After twelve months of this, you'd end up with about $2.61.

What if they compounded daily? You'd get a tiny bit of interest each day, 365 times. That gives you roughly $2.71. Weekly? Hourly? Every second?

Here's the fascinating part: as you compound more and more frequently, the final amount doesn't grow without bound. It approaches a limit. That limit is e. No matter how frequently you compound, you can never exceed e dollars from that initial investment.

Mathematically, e is the limit of the expression one plus one over n, all raised to the n-th power, as n grows infinitely large. It's approximately 2.71828182845904523536... The digits go on forever without repeating, because e is irrational.

A Function That Converts Addition to Multiplication

The exponential function has an extraordinary algebraic property: it transforms sums into products.

If you take e to the power of x plus y, that equals e to the x multiplied by e to the y. Adding the inputs corresponds to multiplying the outputs. This might seem like a trivial consequence of how exponents work, but its implications are profound.

Consider what this means in reverse. The inverse of the exponential function is called the natural logarithm. If the exponential function converts sums to products, then the natural logarithm converts products to sums. The logarithm of x times y equals the logarithm of x plus the logarithm of y.

This is why logarithms were historically so important for computation. Before calculators, multiplying large numbers was tedious and error-prone. But if you had a table of logarithms, you could look up the logs of your numbers, add them together, and then look up what number corresponded to that sum. Addition is much easier than multiplication, so logarithms made difficult calculations tractable.

Slide rules work on this principle. The spacing between numbers on a slide rule corresponds to their logarithms, so when you physically add distances by sliding the rule, you're actually multiplying numbers.

The Power Series

There's another way to define the exponential function that seems completely unrelated to everything I've described. You can write it as an infinite sum:

e to the x equals 1 plus x plus x squared divided by 2 factorial plus x cubed divided by 3 factorial plus x to the fourth divided by 4 factorial, and so on forever.

Here, factorial means multiplying all the positive integers up to that number. So 4 factorial is 4 times 3 times 2 times 1, which equals 24. And by convention, 0 factorial equals 1.

This series converges for every possible value of x, whether x is positive, negative, tiny, or enormous. Plug in any number, add up enough terms, and you'll get an increasingly accurate approximation of e to the x.

What's remarkable is that this infinite series definition gives you the exact same function as the "equals its own derivative" definition. Two completely different approaches to defining a function, and they produce identical results. This is a hint that we've stumbled onto something fundamental about the mathematical universe.

The power series also makes it obvious why the derivative works out so nicely. If you take the derivative of each term in that series, the x terms become 1s, the x-squared terms become 2x (which cancels with the 2 in the denominator), the x-cubed terms become 3x-squared (which cancels with the 3 in 3 factorial), and so on. Everything shifts down by one degree, but the factorial in the denominator perfectly compensates. You get the same series back.

Extending to Complex Numbers

So far, I've been talking about raising e to real number powers. But mathematicians being mathematicians, they wondered: what happens if we try to raise e to imaginary or complex number powers?

An imaginary number is a real number multiplied by i, where i is defined as the square root of negative one. No real number squares to give a negative result, so i is genuinely something new. A complex number is the sum of a real number and an imaginary number, like 3 plus 2i.

What could it possibly mean to raise e to the power of i? The power series gives us an answer. Just plug i into the series and see what happens:

e to the i equals 1 plus i plus i-squared over 2 factorial plus i-cubed over 3 factorial plus i-to-the-fourth over 4 factorial, and so on.

Since i-squared equals negative 1, i-cubed equals negative i, i-to-the-fourth equals 1, and the pattern repeats every four terms, you can separate this series into its real and imaginary parts. The real parts form the series for cosine. The imaginary parts form the series for sine.

This leads to Euler's formula, one of the most beautiful equations in mathematics: e to the i-theta equals cosine of theta plus i times sine of theta.

Read that again. The exponential function, when given an imaginary input, produces outputs involving the circular functions sine and cosine. Exponentials and trigonometry, two subjects that seem completely unrelated in high school, are secretly the same thing when you extend to complex numbers.

The Most Beautiful Equation

Euler's formula has a special case that mathematicians often call the most beautiful equation ever written. Set theta equal to pi:

e to the i-pi equals negative 1.

Or, rearranging:

e to the i-pi plus 1 equals 0.

This single equation connects five of the most important numbers in mathematics. There's e, the base of natural logarithms. There's i, the foundation of complex numbers. There's pi, the ratio of a circle's circumference to its diameter. There's 1, the multiplicative identity. And there's 0, the additive identity. All of them, linked by one concise relationship.

It's not just aesthetically pleasing. It's a hint that these fundamental constants aren't really separate things. They're different faces of the same underlying mathematical structure.

The Graph

Picture the graph of e to the x in your mind. When x equals 0, e to the 0 equals 1, so the graph passes through the point 0, 1. As x increases, the function grows. Slowly at first, then faster and faster, shooting upward in that characteristic exponential curve.

For negative values of x, the function shrinks toward zero but never reaches it. The graph approaches the horizontal axis asymptotically, getting arbitrarily close but never touching. The function is always positive, never zero, never negative.

At every point on this curve, the slope equals the height. Where the function equals 1, the slope is 1. Where it equals 10, the slope is 10. This geometric interpretation of the "equals its own derivative" property is visible in the shape of the curve itself.

The exponential function grows faster than any polynomial. No matter how high a power of x you choose—x squared, x to the hundredth, x to the millionth—eventually e to the x will overtake it and stay above it forever. This is why exponential growth is so powerful and, in contexts like population or resource consumption, so dangerous.

Exponential Functions with Other Bases

I've focused on e because it's mathematically natural, but you can define exponential functions with other bases too. The function 2 to the x doubles every time x increases by 1. The function 10 to the x multiplies by 10 for each unit increase in x.

These can all be rewritten in terms of e. The function 2 to the x equals e to the power of x times the natural logarithm of 2. Any exponential function with any positive base can be expressed using the natural exponential function.

This is why e is called the natural base. It's not that other bases are wrong or unnatural. But e is the one that makes the calculus simplest. With base e, the derivative of the exponential function is itself. With any other base, you pick up an extra constant factor.

Beyond Numbers

Here's where things get really interesting. The exponential function can be generalized beyond numbers entirely.

Matrices, the rectangular arrays of numbers that show up in linear algebra, can be exponentiated. You define e to the power of a matrix using the same power series: the identity matrix plus the matrix plus the matrix squared divided by 2 factorial, and so on. This matrix exponential is crucial in solving systems of differential equations and understanding continuous symmetry transformations.

In physics, especially quantum mechanics, operators representing physical quantities get exponentiated to describe how systems evolve over time. The exponential map in Lie group theory connects infinitesimal symmetries to finite transformations.

The same basic idea—the function that equals its own derivative—keeps showing up at higher and higher levels of abstraction. What starts as a curiosity about real numbers becomes a organizing principle for much of advanced mathematics.

Existence and Uniqueness

I claimed at the beginning that there's exactly one function that equals its own derivative and passes through the point 0, 1. Let me explain why that must be true.

Suppose there were two such functions, call them f and g. Both equal their own derivatives. Both output 1 when given input 0. Consider the ratio f divided by g. Using the quotient rule from calculus, the derivative of this ratio turns out to be zero everywhere. A function whose derivative is always zero must be constant. And since f of 0 divided by g of 0 equals 1 divided by 1 equals 1, that constant must be 1. So f equals g everywhere.

That's uniqueness. Existence we get either from the power series, which you can verify term by term converges and has the required properties, or from the theory of differential equations, which guarantees that equations of this type always have solutions.

Why It Matters

The exponential function isn't just a mathematical curiosity. It's a tool that describes how the physical world actually works.

Radioactive decay follows exponential curves. The atmospheric pressure decreases exponentially with altitude. The charge on a capacitor draining through a resistor follows an exponential. The intensity of light passing through an absorbing medium decreases exponentially with distance.

In finance, compound interest generates exponential growth. In biology, unchecked population growth is exponential. In epidemiology, the early stages of disease spread follow exponential curves—something the whole world learned viscerally during recent pandemics.

Understanding exponential behavior is increasingly important for navigating the modern world. When something grows by a fixed percentage per unit time—whether it's a retirement account, a viral video's view count, or an infectious disease—you're dealing with exponential change. The mathematics is always the same, always tracing back to this one remarkable function that equals its own rate of change.

The exponential function is where pure mathematics meets physical reality. It's an abstraction that describes how change actually works in the universe. And at its heart is a beautifully simple defining property: the only function whose slope at every point exactly matches its height.