Folium of Descartes & Implicit Differentiation | History, Derivative, Tangent Line - Calculus

Take the review quiz on Implicit Differentiation after the lesson: https://1drv.ms/w/c/2713457aa143ce7f/IQC-1nsh-1_PSaL-uQSBMOpEAcR7zYqt2LKnlpYgEbBvQvU?e=0gdtOJ

Folium of Descartes & Implicit Differentiation | History, Derivative, Tangent Line at (3,3)
In this lesson from the MathSupport channel, @mathsupportnow , explore one of the most beautiful and historically important curves in mathematics — the Folium of Descartes — and use it to demonstrate the full power of implicit differentiation.
You’ll learn the history behind the curve, why Fermat and Descartes debated its tangent, and how implicit differentiation provides the correct slope for relations that are not explicitly solved for y.

This is a clear, step‑by‑step walkthrough perfect for Pre‑Calculus 12, Calculus 12, and first‑year university calculus students.

✅What You’ll Learn in This Video
🔹 1. The History Behind Implicit Differentiation
You’ll learn:
Why Fermat is credited with developing implicit differentiation
Why Descartes challenged Fermat using the folium
How the curve became a proof that implicit differentiation works for any implicit relation

🔹 2. The Folium of Descartes — Geometry & Behavior
You’ll explore: The relation 𝑥 ^ 3 + 𝑦 ^ 3 = 3 𝑎𝑥𝑦
Symmetry across the line 𝑦 = 𝑥
How the parameter a changes the size of the “leaf”
The diagonal asymptote 𝑥+𝑦+𝑎=0
How the graph flips when a becomes negative

🔹 3. Differentiating the Folium Implicitly
You’ll learn how to:
Differentiate both sides with respect to x and follow the chain Rule

🔹 4. Finding the Tangent Line at (3,3)
You’ll learn how to: Substitute 𝑥=3, 𝑦=3
into the derivative , Compute the slope 𝑚=−1
Use point‑slope form to write the tangent line
Final tangent line:𝑥 + 𝑦 = 6

00:00 – Introduction
00:07 – Why implicit differentiation matters
00:16 – Fermat vs Descartes (historical context)
01:37 – Graphing the folium on Desmos
02:29 – Symmetry & asymptotes
03:49 – Implicit vs explicit equations
04:15 – Setting up the derivative
05:20 – Differentiating both sides
06:49 – Simplifying and solving for y′
07:40 – Final derivative
08:00 – Tangent line at (3,3)
09:23 – Final equation of tangent line
09:47 – Historical conclusion
10:05 – Outro & next lessons

This video is ideal for:
🎯 Pre‑Calculus 12
🎯 Calculus 12
🎯 First‑year university calculus
🎯 Students learning implicit differentiation
🎯 Anyone interested in math history + beautiful curves

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Видео Folium of Descartes & Implicit Differentiation | History, Derivative, Tangent Line - Calculus канала Mathsupport