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Introduction to Derivatives

A visual guide to rates of change and tangent lines.

INI. Newton8 min readCalculus

1def derivative(f, x, h=1e-7):

2 return (f(x + h) - f(x)) / h

DEFINITION 1

The derivative measures how quickly a function's output changes relative to its input.

EXAMPLE 1

Differentiating a quadratic gives a linear function — the rate of change grows steadily.

NOTE

A function can only be differentiated where it is smooth and continuous.

Function	Derivative	Rule
x²	2x	Power
sin x	cos x	Trig
eˣ	eˣ	Exponential

Which rule applies to differentiating x⁵?

APower rule

BChain rule

CProduct rule

Format

Every concept gets two zones

Each scene pairs a body — the narrative — with a detail zone: the diagram, code, or equation that shows it. Every idea has a place to be explained and a place to be seen.

Interactive

Change a value. Watch everything move.

Parametric diagrams respond to variables in real time. Adjust a parameter and every equation, graph, and scene that depends on it updates live.

Which rule applies to differentiating $x^5$ ?

APower rule

BChain rule

CProduct rule

Assessment

Test yourself without leaving

Multichoice, Wordbank, and Math Response — which evaluates actual algebraic expressions. The test lives inside the document.

e^{i\pi}+1=0

a^2+b^2=c^2

\ln e=1

\frac{d}{dx}x^n=nx^{n-1}

\int e^x\,dx=e^x+C

\sin^2\theta+\cos^2\theta=1

i^2=-1

\nabla^2\phi=0

\frac{d}{dx}\ln x=\frac{1}{x}

e=\lim_{n\to\infty}\!\left(1+\tfrac{1}{n}\right)^n

\det(AB)=\det A\det B

\sum_{n=1}^{\infty}\tfrac{1}{n^2}=\tfrac{\pi^2}{6}

P(A|B)=\frac{P(B|A)P(A)}{P(B)}

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

F=G\frac{m_1 m_2}{r^2}

\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}

\frac{\partial^2 u}{\partial t^2}=c^2\nabla^2 u

e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}

F=ma

\nabla\times B=\mu_0 J

LaTeX

Mathematics rendered properly

Inline and display blocks. Fractions, integrals, summations, matrices — exactly as they should look. Not screenshots. Not approximations.

derivative.py

1def derivative(f, x):

2 h = 1e-7

4 return (f(x + h) − f(x)) / h

6x = lambda t: t**2

7print(derivative(x, 3))

8# ≈ 6.000000008

10g = lambda t: t**3

11print(derivative(g, 2))

12# ≈ 12.000000016

14def d2(f, x):

15 inner = lambda t: derivative(f, t)

16 return derivative(inner, x)

Code

Down to the syntax

Syntax highlighting and editable interactive mode. Diff annotations and line highlighting. Code that's meant to be read and understood.

Plugins

Extend with custom interactive elements

Developers build plugins using the Element SDK — custom interactive elements that run sandboxed inside internotes. Physics simulations, data visualisers, domain-specific tools. Import them with @plugin and they become first-class elements.

DEFINITION 1

EXAMPLE 1

NOTE

TASK 1

Structure

Callouts that signal before you read

Definition, Example, Note, Task — semantic callout types that communicate what kind of content follows before a word is read.

MathematicsPhysicsChemistryComputer ScienceLiteratureHistoryBiologyEconomicsPhilosophyStatisticsEngineeringMusic

Library

Every discipline. Built by people who know it.

Mathematics, Physics, Chemistry, Computer Science, Literature, History — the library spans the full breadth of academic knowledge.

Why do we set the derivative to zero to find the minimum?

Think about what the slope is doing at the very bottom of a curve — what value does it take there?

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Frequently Asked Questions

An interactive educational document that combines prose, equations, code, diagrams, and live scenes in one place. Unlike a static page, content responds when you reach in and interact with it — adjust a parameter and every element that depends on it updates live. Assessment questions live inside the document too, so you can test your understanding without switching tools or losing your place.

A PDF is fixed — the content is the same no matter what you do. An internote responds: change a parameter and every element that depends on it updates in real time. It can also contain assessment questions that let you test your understanding inside the document, without switching tools. The content is designed to be interacted with, not just read.

Internotes combine the kinds of content that learning actually needs — prose, mathematics, code, and diagrams — with interactive elements that respond when you engage with them, and assessment questions that let you test yourself without leaving the page. The format is designed to put each type of content where it explains best, rather than forcing everything into a single text block.

The library spans the breadth of academic knowledge — mathematics, physics, chemistry, computer science, literature, history, biology, economics, and more. Some internotes are concise five-minute explainers; others are full chapter-length treatments of a topic. The depth and scope varies by subject and author.

Plugins are custom interactive elements that can be embedded inside an internote — physics simulations, data visualisers, chemistry tools, and anything else built for a specific domain. They run safely inside the document and can connect to the rest of the internote, updating live when parameters change.

Yes. Public internotes are fully readable without signing in — no account is required to browse, read, or interact with the content. An account is only needed to create internotes, save your progress across sessions, or use Ask Internote.

Function

Derivative

Rule

x²

Power

sin x

cos x

Trig

eˣ

Exponential

Frequently Asked Questions