Foundations — Derivatives of ln x and logₐ(x)

This page assumes you have seen nothing. Before we can even talk about the slope of $\ln x$, we must earn every mark on the page: what a power is, what a log undoes, what a limit is, what $e$ is, what a slope is, and what the strange squiggle $\frac{d}{dx}$ means. We build them in order — each block leans only on the ones above it.

1. Powers — the starting brick

Picture it. $2^3$ is three copies of $2$ stacked as a product: $2\times 2\times 2=8$. As the exponent climbs, the value shoots upward faster and faster.

Read the figure. Each bar is $2^n$ for $n=0,1,2,3,4,5$. Notice the leftmost bar is height $1$ (that is $2^0$, explained just below) and each bar to the right is double the one before — that doubling is what "multiply by another copy of $2$" looks like, and the magenta arrow marks how fast it runs away.

Why the topic needs this. Everything about logs is written on top of powers. You cannot ask "what power gives me this?" until you know what a power is — for every kind of power, whole, zero, negative or fractional. See Derivative of e^x and a^x for where these grow into functions.

2. The logarithm — asking the exponent backwards

So a log is not a new machine — it is the undo button for a power. The two lines below say the same thing read left-to-right vs right-to-left:

$a^y=x⟺y={\log }_{a}x.$

Picture it. A power takes an exponent $y$ and produces a value $x$ that grows explosively. The log is the mirror image: it takes the big value $x$ and reads off the small exponent $y$.

Read the figure. The orange curve is $y=2^x$; feed it the exponent $3$ (bottom axis) and it outputs $8$ (marked orange dot). The magenta curve is its undo, $y={\log }_{2}x$; feed it $8$ and it returns $3$ (magenta dot). The two curves are exact reflections across the dashed navy line $y=x$ — that reflection is what "inverse / undo" looks like geometrically.

Also we insist $a>0$ and $a\ne 1$: if $a=1$ then $1^y=1$ always, so it could never equal any other $x$ — the question has no answer.

3. Limits — "the value we head towards"

Before we can build $e$ or a slope, we need one honest tool for "getting closer and closer without necessarily arriving." (We already used it once above, to define $a^{\sqrt{2}}$.)

Why we need this. Three ideas — giving $a^x$ a meaning for irrational $x$, the number $e$, and the slope of a curve — all ask "what happens as something shrinks/grows, if I can't just plug in the endpoint?" The limit is the safe way to answer that. It underpins Standard Limit (1+t)^{1/t} → e.

4. The number $e$ — nature's favourite base

Where does it come from? Watch a bank that pays interest and re-invests it more and more often. Start with \1$.Interestappliedonceattheend:$(1+1)^1 = 2$.Splitintotwohalf-paymentscompounded:$(1+\tfrac12)^2 = 2.25$.Into$n$pieces:$\left(1+\tfrac1n\right)^n$.Push$n$everlarger—usingthelimitfrom\S 3—andthenumberstopsclimbing;itsettleson$e$:

$e={\lim }_{n\to \infty }{\left(1+\frac{1}{n}\right)}^n,\text{equivalently}{\lim }_{t\to 0}(1+t{)}^{1/t}=e.$

Why the second form? Set $t=1/n$. Then "$n$ huge" becomes "$t$ tiny," i.e. $n\to \infty$ turns into $t\to 0$ — the very meaning of $\lim$ we just defined. See Standard Limit (1+t)^{1/t} → e.

Read the figure. Each violet dot is $(1+1/n{)}^n$ for a growing $n$ (horizontal axis, log-spaced). Early dots climb fast, later dots barely move, and all of them press up against the dashed magenta line at height $e\approx 2.718$. That flattening-onto-a-line is exactly what "$\lim =e$" looks like.

5. Slope — how steep is the hill?

Picture it. Two points on a line make a right triangle: the horizontal side is the run, the vertical side is the rise. Slope is how tall that triangle is per unit width.

Read the figure. The orange curve is $y=\ln x$. The dashed magenta line just kisses it at $x=1$; the little navy triangle shows a run along the bottom and the matching rise up the side — their ratio is the slope. Notice the violet note far right: the curve there is nearly flat, so its rise-per-run (slope) is tiny.

But $\ln x$ is a curved hill — its steepness is different at every point. So we need one more idea: the slope at a single point.

How we compute it. Let $h$ stand for a small change in $x$ — how far we step sideways from $x$ to $x+h$. Then the rise is $f(x+h)-f(x)$ and the run is $h$, so the rise-over-run of that little step is $\frac{f(x+h)-f(x)}{h}$. Now squeeze the run down to nothing using the limit from §3:

$f^{\prime }(x)=\frac{dy}{dx}={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}.$

Why the limit? If we used a run of exactly zero we would divide by zero — forbidden. So we watch what the rise-over-run approaches as the step $h$ shrinks. That "approaches" is precisely what ${\lim }_{h\to 0}$ captures.

6. Two facts we must earn: $\frac{d}{dx}{e}^x=e^x$, then $\frac{d}{dx}\ln x$

The parent topic hinges on the slope of $\ln x$. To get it honestly we first show why the exponential $e^x$ is its own slope, using only the limit tool and the definition of $e$.

6a. Why $e^x$ differentiates to itself

Apply the derivative-as-limit to $f(x)=e^x$:

$\frac{d}{dx}{e}^x={\lim }_{h\to 0}\frac{e^{x+h}-e^x}{h}.$

What we just did: wrote out the rise-over-run for $e^x$. Why: it is the only definition of slope we have.

Use the power-law $e^{x+h}=e^x\cdot e^h$ (from §1) and pull the common $e^x$ out — it does not depend on $h$:

$={\lim }_{h\to 0}\frac{e^x{e}^h-e^x}{h}=e^x\cdot {\lim }_{h\to 0}\frac{e^h-1}{h}.$

Why this step: factoring $e^x$ separates a constant (as far as the limit is concerned) from the one piece that still moves.

Now the whole question is: what is $\underset{h\to 0}{\lim }\frac{e^h-1}{h}$? Watch what $e^h-1$ looks like for tiny $h$. From $e={\lim }_{t\to 0}(1+t{)}^{1/t}$ (§4), for a tiny $t$ we have $e^t\approx 1+t$, so $e^t-1\approx t$, hence $\frac{e^t-1}{t}\to 1$:

${\lim }_{h\to 0}\frac{e^h-1}{h}=1.$

What it looks like: the curve $y=e^x$ crosses the $y$-axis at height $1$ with slope exactly $1$ — that "slope $=1$ at the start" is where the $1$ comes from. Substituting back:

$\boxed{\frac{d}{dx}{e}^x=e^x\cdot 1=e^x.}$

So $e^x$ is the one function whose slope at every point equals its own height. That is precisely the special property we announced for the base $e$ in §4 — now proved. (This is developed further in Derivative of e^x and a^x.)

6b. From that, the slope of $\ln x$ — every algebra step shown

We want $\frac{d}{dx}\ln x$. Start from the defining relation, not a formula. Let $y=\ln x$, which by §2 means

$e^y=x.$

What we just did: rewrote $\ln$ as the exponent that lands on $x$. Why: we already know how to differentiate $e^{\text{something}}$, but not $\ln$ directly.

Now differentiate both sides with respect to $x$. Left side is $e^y$ where $y$ is itself a function of $x$; right side is just $x$:

$\frac{d}{dx}\left(e^y\right)=\frac{d}{dx}(x).$

The right side is $1$ (a slope-$1$ line). The left side needs the chain rule (§7): the outside is $e^{(\cdot )}$, whose derivative is itself by §6a, times the derivative of the inside $y$:

$e^y\cdot \frac{dy}{dx}=1.$

What we just did: "outside derivative times inside derivative." Why: $y$ depends on $x$, so a bare $e^y$ hides an inner change we must account for. What it looks like: the height $e^y$ moves both because $e$ is steep and because $y$ itself is shifting.

Solve for $\frac{dy}{dx}$ by dividing both sides by $e^y$, then replace $e^y$ with $x$ (they are equal, from the top line):

$\frac{dy}{dx}=\frac{1}{e^y}=\frac{1}{x}.$

$\boxed{\frac{d}{dx}\ln x=\frac{1}{x},x>0.}$

7. Two tools that ride on top

Why the topic needs it. Real problems are $\ln (3x^2+1)$, not bare $\ln x$. The inside $u=3x^2+1$ must be accounted for, giving $\frac{u^{\prime }}{u}$. We already used exactly this rule in §6b to differentiate $e^y$.

Why the topic needs it. In change-of-base, ${\log }_{a}x=\frac{\ln x}{\ln a}$, the denominator $\ln a$ is a fixed number — it factors out, leaving $\frac{1}{\ln a}\cdot \frac{1}{x}=\frac{1}{x\ln a}$. This is the whole derivation of $\frac{d}{dx}{\log }_{a}x$ (see Logarithm Laws and Logarithmic Differentiation for the deeper uses).

8. How it all feeds the topic

Equipment checklist

Cover the right side and test yourself. If any answer is a surprise, re-read that section before the parent note.