Foundations — Derivatives and rules (product, quotient, chain)

Before you can read "$f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$" without flinching, every mark on that line must mean something concrete to you. This page builds each symbol from nothing, in the order they lean on each other. If the parent note used it, we define it here first.

1. Function and the notation $f(x)$

• Plain words: $f$ is the name of the machine, $x$ is what you put in, $f(x)$ is what comes out.
• The picture: a graph. The horizontal axis is the input $x$; the vertical axis is the output $f(x)$. Each input picks one point on the curve.
• Why the topic needs it: derivatives describe how a machine responds to its input. No machine, nothing to differentiate.

Notice $f(x)=x^2$ in the figure: input $2$ gives output $4$, input $-1$ gives output $1$. The curve is just the collection of all such (input, output) dots joined up.

2. The variable $x$ and a "nudge" $h$

• Plain words: $x$ is where you stand; $x+h$ is where you stand after taking a small step of size $h$.
• The picture: two vertical lines on the graph, one at $x$, one a little further at $x+h$. The gap between them is $h$.
• Why the topic needs it: to measure change you must compare two nearby inputs. $h$ is that gap. The whole idea of a derivative is "let this gap shrink to nothing."

3. $\Delta$ (change) and slope = rise over run

• The picture: a straight line touching the curve at two points — a secant line. Its tilt is the slope. See the yellow line below.
• Why the topic needs it: the derivative is a slope. But the fraction above is only an average slope across the whole step $h$ — the average steepness of a hill from bottom to top, not the steepness right where you stand.

4. The limit ${\lim }_{h\to 0}$

• Plain words: "where is this heading as the step shrinks to nothing?"
• The picture: the two vertical lines slide together. The secant line pivots until it just grazes the curve at a single point — that grazing line is the tangent line.
• Why this tool and not just "plug in $h=0$"? If you set $h=0$ directly in $\frac{f(x+h)-f(x)}{h}$ you get $\frac{0}{0}$ — undefined nonsense. The limit is the honest way to ask "what would the slope be if the step were infinitely small" while dodging the divide-by-zero. This is the whole reason Limits and Continuity comes before derivatives.

Watch the sequence: as $h$ goes $1\to 0.5\to 0.1$, the yellow secant swings toward the red tangent. The tangent's slope is the derivative.

5. The derivative and its symbols $f^{\prime }(x)$ and $\frac{dy}{dx}$

Two names for the same thing:

• The $d$ is $\Delta$ after the limit: an infinitely small change. $\frac{dy}{dx}$ is literally rise-over-run once the run is infinitesimal.
• Why two notations? The chain rule looks like a fraction cancelling ($\frac{dy}{du}\cdot \frac{du}{dx}$) in Leibniz form $\frac{dy}{dx}$, so that notation is handy. The prime form is compact for stating rules. You will meet both, so recognise both.
• Why the topic needs it: this single number — "output-change per input-nudge" — is what training a model uses to decide which way to push each weight. See Gradient and Partial Derivatives and Gradient Descent.

6. Building blocks the rules assume

The three combination rules (the parent topic) assume you already recognise these pieces.

• Why rename? Writing $\frac{d}{dx}[x^2\sin x]$ in full is bulky. Calling $u=x^2$, $v=\sin x$ lets the rule read $u^{\prime }v+u{v}^{\prime }$ — a template you fill in.

• The picture: for $x^2$, the graph is a bowl; its slope $2x$ is negative on the left, zero at the bottom, positive on the right. See Power Rule and Basic Derivatives.
• Why the topic needs it: powers are the simplest non-trivial machines; every worked example (like $(3x^2+1{)}^5$) is built from them.

• Why this tool? The exponential is the natural currency of growth/decay and shows up inside the sigmoid. Its self-slope property is what makes the sigmoid derivative come out so clean. This feeds Activation Functions.

• The picture: a flat-then-rising-then-flat "S". Below.
• Why the topic needs it: it's the parent note's showcase example combining all three rules; recognising its shape tells you what its derivative should look like (a bump, biggest in the middle).

7. How these feed the topic

Read it top-down: a function gives us something to nudge; the nudge and delta give a secant slope; the limit turns that average slope into the exact derivative; power rule and the exponential are the raw functions; together they power the three rules, which produce things like the sigmoid derivative used in Backpropagation.

Equipment checklist

Self-test: can you answer each before revealing?

Connections

• Limits and Continuity — where $\lim$ is built properly.
• Power Rule and Basic Derivatives — the raw building blocks.
• Activation Functions — where the sigmoid lives.
• Gradient and Partial Derivatives, Gradient Descent, Backpropagation — where derivatives get used.