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, The folium of Descartes is an algebraic
curve carved out by a certain equation. By

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which equation? This equation, x cubed
plus y cubed minus 3axy equals 0. It's the

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points on the plane that satisfy this
equation.

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So, what's a folium? Well, folium is just
a Latin word for leaf, you know, the sorts

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of things that grow on trees. So, where's
the leaf? Well, here's the leaf. I've

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plotted the points on the plane that
satisfy x cubed plus y cubed minus 9xy

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equals 0. And this is the curve that I
get, and you can see it looks kind of like

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a leaf. This is not the graph of a
function, it's really a relation. x cubed

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plus y cubed minus 9xy is a polynomial in
two variables, in both x and y, in both. I

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can't solve for x in terms of y. Look,
this graph fails the vertical line test.

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For a given value of x, there's
potentially multiple values of y which

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will satisfy this equation. So, what's the
point of all these? Well, once upon a

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time, Descartes challenged Fermat to find
the tangent line to this folium. And

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Descartes couldn't do it but Fermat could.
And now, so can you. And you can do it

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with implicit differentiation. So, let's
use implicit differentiation on this,

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thinking of y secretly as a function of x.
So, the derivative of x cubed is 3x

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squared. The derivative of y cubed, well,
that's 3y squared times dy dx, that's

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really the Chain rule in action, minus,
now it's got to differentiate this. It

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will be 9 times the derivative x, which is
1y minus 9x times the derivative of y,

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which is dy dx, and that's equal to 0.
Alright, now I can rearrange this, the

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things with the dy dx, and the things
without the dy dx, and you gather it

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together. So, 3x squared minus 9y plus,
and the things with the dy dx term, 3y

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squared minus 9x dy dx equals 0. Now, I'm
going to subtract this from both sides.

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So, I'll have 3y squared minus 9x times dy
dx equals minus 3x squared plus 9y. And

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I'm going to divide both sides by this, so
I'll have dy dx equals minus 3x squared

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plus 9y over 3y squared minus 9x. And note
that we're calculating dy dx but the

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answer involves both x and y. And you can
see, it's really working. I can pick a

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point on this curve like a point 4, 2
satisfies this equation. Then, I can ask

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what's the slope of the tangent line to
the curve through the point 4, 2? When I

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go back to our calculation of the
derivative and if I plug in 4 for x and 2

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for y, I get that the derivative is 4/5.
And indeed, I mean, this graph is somewhat

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stretched, but, you know, yeah, I mean
that doesn't look terribly unreasonable

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for the slope of this line. Problems like
this one, which once stumped the smartest

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people on earth can now be answered by
you, by me, by lots and lots of people.

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Calculus is part of a human tradition of
making not just impossible things

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possible, but things that were once really
hard much easier.

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Well, in any case, there's plenty more
questions that you can just ask about

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different kinds of curves besides this
folium of Descartes. You can write down

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some polynomial with x's and y's, like y
squared minus x cubed minus 3x squared

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equals = 0 and then you can ask about the
points, the x comma y's that satisfy this

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equation.
And if you want to know the slope of the

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tangent line, use implicit
differentiation. The trick is just to use

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the Chain rule and to treat y as a
function of x.