Calculus I · The Function That Is Its Own Slope

Hunting for the base that leaves the axis at slope 1

We want the base $a$ where the curve crosses the $y$-axis at slope exactly $1$ — where $M(a) = 1$. Let us just measure $M(a)$ for a few bases by taking very small $h$ in $\dfrac{a^h - 1}{h}$. (This is the kind of thing a calculator, or the notebook for this chapter, does in a heartbeat. Try $h = 0.0001$.)

For $a = 2$ the curve leaves the axis too gently — slope only $0.693$. For $a = 3$ it leaves a little too steeply — slope $1.099$. So somewhere between $2$ and $3$ there is a base where the slope is exactly $1$. (If you want this argument made airtight: $M(a)$ slides smoothly upward as $a$ grows, it is below $1$ at $a=2$ and above $1$ at $a=3$, so it must hit $1$ somewhere in between. We will sharpen “slides smoothly” later, but the picture is honest.)

That special base gets its own letter. We call it $e$:

$$e \approx 2.71828\ldots$$

and it is defined to be the base for which $M(e) = 1$. The instant we say that, the boxed rule gives us our prize:

$$\frac{d}{dx}e^{x} = M(e)\cdot e^{x} = 1\cdot e^{x} = e^{x}.$$

$$\boxed{\;\frac{d}{dx}\,e^{x} = e^{x}\;}$$

This is the function that is its own slope. We did not pull it out of a hat. We went looking for a function equal to its own derivative, discovered that every exponential is almost one (off only by a constant $M(a)$), and then picked the one base that makes the constant vanish. The number $e$ is exactly the price of that wish.

(Figure 5.2: $y = e^x$ with tangent lines drawn at several points. At every point, the steepness of the tangent equals the height of the curve. Where the curve is at height $5$, the slope is $5$. That is what “its own derivative” looks like.)

A promise about $e$, to be kept in Chapter 1’s other form

Back in Chapter 1, when we played with sequences, we watched the quantity $\left(1 + \frac{1}{n}\right)^n$ creep upward as $n$ grew — $2$, then $2.25$, then $2.4883\ldots$, settling toward $2.718\ldots$. That was the same number $e$. It is not obvious that “the base whose curve leaves the axis at slope $1$” and “the number that $\left(1+\frac1n\right)^n$ sneaks up on” are the same number. They are. This is the loan I warned you about. For now, take it as a promise backed by the matching decimals; in Chapter 10, with integrals in hand, we will prove the two descriptions of $e$ are genuinely one number.

Toolkit: Differentiating exponentials

• $\displaystyle \frac{d}{dx}\,e^{x} = e^{x}$ — the headline. The function is its own derivative.
• $\displaystyle \frac{d}{dx}\,e^{kx} = k\,e^{kx}$ — by the chain rule from Chapter 4 (outer function $e^{(\cdot)}$, inner function $kx$, inner derivative $k$).
• $\displaystyle \frac{d}{dx}\,a^{x} = (\ln a)\,a^{x}$ — the mystery constant $M(a)$ turns out to be $\ln a$. We prove this two pages from now.

The middle line is the one you will use most. The constant $k$ being positive means growth; $k$ negative means decay. That single sign is the difference between rabbits and radioactivity.

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