In this video I go over another example on implicit differentiation and now look at the famous cubic function, the Folium of Descartes. I also go over some very interesting historical facts in the history of math so make sure you watch this video!
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Implicit Differentiation: Example on the Folium of Descartes
Folium of Descartes
x3 + y3 = 3axy
First proposed by the French mathematician René Descartes in 1638.
Its claim to fame lies in an incident in the development of calculus. Descartes challenged Pierre de Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines. Fermat solved the problem easily, something Descartes was unable to do. Since the invention of calculus, the slope of the tangent line can be found easily using implicit differentiation. - Wikipedia
Example
If a = 2
(a) Find y’
(b) Find equation of tangent line at the point (3, 3)
(c) At what points on the curve is the tangent line horizontal?
Solution
Explicit Solution of the Folium
Implicit Differentiation makes finding derivatives very easy without solving explicitly for f(x).
There is a formula similar to the Quadratic Formula but for Cubic Functions (but much more complex). We can solve for the 3 functions of the Folium of Descartes:
Interesting Historical Facts
The Norwegian mathematician Niels Abel proved in 1824 that no general formula can be given for the roots of a fifth-degree equation in terms of radicals (i.e. square roots, cube roots, etc.).
Later the French mathematician Evariste Galois proved that it is impossible to find a general formula for the roots of an n-th degree equation (in terms of algebraic operations on the coefficients) if n is any integer larger than 4.
Moreover, implicit differentiation works just as easily for equations such as:
y5 + 3x2y2 + 5x4 =12
for which it is impossible to find a similar expression for y in terms of x.