Calculus I · The Function That Is Its Own Slope

Every exponential is already proportional to itself

Start with a plain exponential function, $f(x) = a^x$, where $a$ is some fixed positive number — maybe $2$, maybe $10$. You met these in Book 1. Let us just take its derivative from scratch, the way we have taken every derivative so far: write down the difference quotient and see what happens.

$$\frac{f(x+h) - f(x)}{h} = \frac{a^{x+h} - a^{x}}{h}.$$

Now use the one exponent rule that does all the work here: $a^{x+h} = a^x \cdot a^h$. Factor the $a^x$ out:

$$\frac{a^{x}a^{h} - a^{x}}{h} = a^{x}\cdot\frac{a^{h} - 1}{h}.$$

Look hard at what just happened, because it is the heart of the chapter. The $a^x$ came out front and did not change as $h$ shrank. The only part that depends on $h$ is the fraction $\dfrac{a^h - 1}{h}$. So when we take the limit as $h \to 0$:

$$\frac{d}{dx}a^{x} = a^{x}\cdot \underbrace{\left(\lim_{h\to 0}\frac{a^{h}-1}{h}\right)}_{\text{a number that depends only on }a}.$$

That underbraced limit is just a number. It does not depend on $x$ at all. Let us give it a name and call it $M(a)$, for the moment, because it is the mystery number attached to the base $a$:

$$\boxed{\;\frac{d}{dx}a^{x} = M(a)\cdot a^{x}\;}\qquad\text{where}\qquad M(a) = \lim_{h\to 0}\frac{a^{h}-1}{h}.$$

Read that boxed line out loud, because it is astonishing and most people never notice it. Every exponential function is already proportional to its own slope. The derivative of $a^x$ is just $a^x$ scaled by some fixed number $M(a)$. We have almost answered our question for free. The only thing standing between us and a function that equals its own derivative is that pesky constant $M(a)$ out front. If only we could find a base $a$ that made $M(a)$ exactly equal to $1$.

One more thing to notice before we hunt for that base. What is $M(a)$, geometrically? Plug $x = 0$ into the boxed rule. Since $a^0 = 1$, the slope of $a^x$ at $x = 0$ is exactly $M(a)\cdot 1 = M(a)$. So:

$M(a)$ is simply the slope of the curve $y = a^x$ as it crosses the $y$-axis.

That gives us something we can actually look at. (Figure 5.1: the curves $y=2^x$, $y=3^x$, and a couple of others, each with its tangent line drawn where it crosses the $y$-axis. The steeper the curve leaves the axis, the bigger $M(a)$.)

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