The Mathematical Origin of the Natural

The exponential function is as fundamental to the study of calculus—if not more so—than the trigonometric functions. While many students are introduced to the constant $e$ as a mere numerical value (2.718…), its true power and “naturalness” are only revealed through the elegant machinery of differentiation. This document traces the pedagogical shortcut from the basic properties of exponents to the inevitable discovery of the natural base.

1. Fundamental Properties of Exponential Functions

An exponential function is defined by a positive base $a>0$ and is governed by three core algebraic rules that serve as the bedrock for our calculus exploration:

1. The Value at the Origin: $a^0=1$.
2. The Sum Rule: $a^{x_1+x_2}=a^{x_1}\cdot a^{x_2}$ (the exponential of a sum is the product of the exponentials).
3. The Power of a Power Rule: $(a^{x_1}{)}^{x_2}=a^{x_1\cdot x_2}$.

While these properties clearly define $a^x$ for rational exponents, the function is extended to all real numbers—including irrationals like $\sqrt{2}$ or $\pi$—by “filling in” the gaps through continuity. In practice, this is exactly what a calculator does: it uses a high-precision decimal expansion to approximate the theoretical value.

2. Deriving the General Derivative of $a^x$

To find the derivative of an arbitrary exponential function $a^x$, we must return to the first principles of the limit definition. One must note that while $x$ is our variable of interest, for the purpose of the limit, $x$ remains fixed while $\Delta x$ is the moving quantity.

$\frac{d}{dx}{a}^x={\lim }_{\Delta x\to 0}\frac{a^{x+\Delta x}-a^x}{\Delta x}$

Applying the sum rule ($a^{x+\Delta x}=a^x{a}^{\Delta x}$), we factor the numerator:

$\frac{d}{dx}{a}^x={\lim }_{\Delta x\to 0}\frac{a^x(a^{\Delta x}-1)}{\Delta x}$

Since $a^x$ does not depend on $\Delta x$, we can factor it out of the limit, revealing a profound result:

$\frac{d}{dx}{a}^x=a^x\left({\lim }_{\Delta x\to 0}\frac{a^{\Delta x}-1}{\Delta x}\right)$

The term in the parentheses is a “mystery number” that depends solely on the base $a$. We define this as $M(a)$, leading to our preliminary derivative formula:

$\frac{d}{dx}{a}^x=M(a)a^x$

3. The Geometric Interpretation of $M(a)$

The value $M(a)$ represents the slope of the function $a^x$ at the specific point $x=0$. This is easily verified: at $x=0$, the derivative $M(a)a^0$ simplifies to $M(a)\cdot 1$.

Key Insight: The beauty of this approach lies in the fact that knowing the slope of an exponential function at the origin (x=0) allows us to sidestep our initial ignorance and calculate the slope at any other point on the curve.

4. The Scaling Argument for the Existence of '$e$'

To determine the identity of $M(a)$, we take a “circuitous” rather than circular route. We seek to define a base $e$ such that the slope at the origin is exactly 1 ($M(e)=1$).

We can guarantee the existence of such a base through a scaling argument. Consider $f(x)=2^x$. If we stretch this function by a factor $k$, we get $f(kx)=2^{kx}$, which is equivalent to a new base $b=2^k$. Applying the chain rule, the derivative of $f(kx)$ is $k\cdot f^{\prime }(kx)$. At the origin ($x=0$), the slope becomes $k\cdot M(2)$.

Because we can adjust the “stretch” factor $k$ to any value, we can effectively “tilt” the slope at the origin to any value we desire. Consequently, there must exist a unique $k$ that scales the slope to exactly 1. We name the resulting base $e$, providing us with our most essential “keeper” formula:

Much like we prefer radians in trigonometry because they make the derivative of $\sin x$ equal to $\cos x$ without “horrible” constant factors, we choose base $e$ to make the derivative of the exponential function equal to itself.

5. The Natural Logarithm and Inverse Differentiation

The natural logarithm ($\ln x$) is the inverse of $e^x$, such that if $w=\ln x$, then $e^w=x$. Its core properties include:

• $\ln (x_1{x}_2)=\ln x_1+\ln x_2$
• $\ln 1=0$
• $\ln e=1$

Using implicit differentiation on the relationship $e^w=x$:

1. $\frac{d}{dx}(e^w)=\frac{d}{dx}(x)$
2. $e^w\cdot \frac{dw}{dx}=1$
3. $\frac{dw}{dx}=\frac{1}{e^w}=\frac{1}{x}$

Thus, we arrive at the companion formula: $\frac{d}{dx}\ln x=\frac{1}{x}$.

6. Identifying $M(a)$ and General Differentiation Methods

We can now identify the mystery number $M(a)$ as simply $\ln a$. This explains why the natural log is a “fundamental constant of nature.” Even if we insisted on using base 10 or base 2, we would still be burdened by the natural log as a scaling factor (e.g., $\frac{d}{dx}10^x=\ln (10)\cdot 10^x$). Using base 10 is as “horrible” in calculus as using degrees in trigonometry; $e$ and radians are the only truly natural choices.

Method One: Conversion to Base $e$

Any base can be converted using the identity $a^x=e^{x\ln a}$.

$\frac{d}{dx}{a}^x=\frac{d}{dx}{e}^{(x\ln a)}=(\ln a)e^{x\ln a}=(\ln a)a^x$

Method Two: Logarithmic Differentiation

This pedagogical shortcut uses the identity $(\ln u{)}^{\prime }=\frac{u^{\prime }}{u}$. It is particularly powerful for functions with “moving bases and moving exponents,” such as $v=x^x$:

1. $\ln v=x\ln x$
2. Differentiate both sides using the product rule: $\frac{v^{\prime }}{v}=\ln x+x(\frac{1}{x})=\ln x+1$
3. Multiply by $v$: $v^{\prime }=x^x(1+\ln x)$

7. Closing the Loop: The Limit Definition of '$e$'

To transform $e$ from a theoretical construct into a calculable constant, we evaluate the limit ${\lim }_{n\to \infty }(1+\frac{1}{n}{)}^n$.

Let $\Delta x=\frac{1}{n}$. We take the natural log of the expression: $n\ln (1+\frac{1}{n})=\frac{\ln (1+\Delta x)}{\Delta x}$. Subtracting $\ln 1$ (which is 0) clarifies the form:

${\lim }_{\Delta x\to 0}\frac{\ln (1+\Delta x)-\ln 1}{\Delta x}$

This is the derivative of $\ln x$ at $x=1$, which we know is $1/1=1$. Crucially, since the limit of the log equals the log of the limit, we have:

$\ln \left({\lim }_{n\to \infty }(1+\frac{1}{n}{)}^n\right)=1⟹{\lim }_{n\to \infty }(1+\frac{1}{n}{)}^n=e^1$

This provides a practical numerical method for approximating $e$ (e.g., $(1+1/100{)}^{100}$), finally connecting our high-level calculus derivations back to the concept of “filling in” by continuity.