Foundations — Derivatives of eˣ and aˣ — proofs

Before you can follow a single line of Derivatives of eˣ and aˣ — proofs, you need to be fluent in the symbols it throws around without pausing. This page builds each one from nothing, in the order they depend on each other. Read top to bottom; nothing here uses a symbol defined lower down.

1. A variable $x$ and a function $f(x)$

The picture: a number line for the inputs $x$ at the bottom, and for every input a height plotted above it. Join all those heights and you get a curve — the graph of the function.

Why the topic needs it: the whole chapter asks "how steep is this curve?", so we first need the curve itself. In our case the machine is $f(x)=a^x$ or $f(x)=e^x$.

2. Powers: $a^x$, the base, and the exponent

For whole numbers this is easy: $2^3=1\cdot 2\cdot 2\cdot 2=8$.

3. The exponent laws — one law does all the work

Why the topic needs it: this single splitting trick is the reason the derivative of an exponential comes back as itself. In the parent proof, $a^{x+h}=a^x{a}^h$ lets the fixed part $a^x$ slide out of the limit, leaving a pure number behind.

4. Slope: rise over run

The picture: pick two points on the line. Draw a horizontal step (the run) and the vertical step it forces (the rise). Their ratio is the slope, and it's the same everywhere on a straight line.

Why the topic needs it: "how steep is the exponential?" is the question "what is its slope?" But a curve's steepness changes from point to point — so we need one more idea.

5. The limit symbol $\underset{h\to 0}{\lim }$

Why the topic needs it: every derivative in the parent note is a limit, and the special constant of §7 is defined by a limit. We introduce it now, before any formula uses it.

6. The tangent line and the derivative $f^{\prime }(x)$

How we compute it (first principles): pick your point $x$, and a nearby point $x+h$ a tiny run $h$ away. The line through those two points (a secant) has slope $\frac{f(x+h)-f(x)}{h}=\frac{\text{rise}}{\text{run}}.$ Now let the run $h$ shrink toward $0$ — using the limit symbol just defined in §5 — the second point slides into the first, and the secant swings until it becomes the tangent. That final slope is the derivative, written $f^{\prime }(x)$.

Why the topic needs it: this formula is the entry gate of the entire proof. Everything the parent note derives starts by feeding $a^x$ into this machine.

7. The slope-at-zero constant $M(a)$, and the number $e$

Why the topic needs it: $e$ is the whole point — it is the base that makes $e^x$ its own derivative.

8. The natural logarithm $\ln a$

Why the topic needs it: the trick $a=e^{\ln a}$ rewrites every exponential in terms of $e$, so the one clean rule ($e^x$ is self-derivative) can crack them all. The leftover factor turns out to be $\ln a$ — the true identity of the mystery constant $M(a)$ from §7.

9. The chain rule (the one composition tool)

Why the topic needs it: after writing $a^x=e^{(\ln a)x}$, the inner function is $u=(\ln a)x$ with slope $u^{\prime }=\ln a$, and the outer is $g(u)=e^u$ with $g^{\prime }(u)=e^u$. The chain rule pulls out that $\ln a$ — delivering $\frac{d}{dx}{a}^x=a^x\ln a$.

How the pieces feed the topic

Read the map like a river flowing downhill: the sources at the top (a function, powers, slope, the limit) merge into the first-principles derivative, which produces the constant $M(a)$; choosing the base that makes $M(a)=1$ defines $e$ and gives the self-derivative on the left branch, while rewriting any base through $\ln a$ and applying the chain rule gives the general rule on the right branch. Both branches land on the two boxed results of the parent note.

Equipment checklist

Test yourself — you are ready for the parent proof only if you can answer every line.

Connections

• Derivatives of eˣ and aˣ — proofs — the parent topic this page prepares you for.
• Limits — first principles definition of derivative — where the derivative formula comes from.
• The number e — definitions — deeper look at $e$.
• Natural Logarithm ln x — the inverse of $e^x$.
• Chain Rule — composition tool used in §3 of the parent.
• Power Rule — the contrasting rule for fixed exponents.
• Exponential Growth and Decay — where self-derivative becomes a model, including decay $0<a<1$.