Tschirnhausen cubic

The Tschirnhausen cubic

As an example of implicit differentiation, we study the Tschirnhausen cubic.

Since each value of x in the interval x > -3 except x = 0 corresponds two different y-values, the cubic does not determine y as a function of x. The colors in the drawing are meant to suggest one way in which we could divide the cubic into two parts, each of which determines y as a function of x in a different way. But this way of dividing the cubic is not the only possible one; to draw this cubic, I instructed my computer algebra system to draw the graphs of the two functions shown here:

Unlike the two functions in the first picture, these two functions are not differentiable at 0; but they have simple expressions in terms of familiar functions. However the cubic is divided, though, if we focus on a sufficiently small part of the curve, except (-3,0) and (0,0), y appears as a differentiable function of x. In this sense, we say that the equation of the cubic determines y "implicitly" as a function of x (as opposed to an "explicit" definition of y as a function of x). Suppose we want to find the derivative dy/dx of the function at one of these many "good" points. For this cubic, since it is possible to express the functions explicitly (as written on the graph), we need not use implicit differentiation. But for many curves explicit solutions are not available, so let us try something else. Granting that y is (locally) a function of x,, both sides of the defining equation y²=x³+3x² are functions of x, indeed, the same function of x because they are equal. So their derivatives are equal also. Now, although we are avoiding the explicit expression of y in terms of x, we can still use the Chain Rule to write an expression for the derivative with respect to x of the left side of the equation, which must equal the derivative of the right side:

2y(dy/dx) = 3x²+6x

Solving this for dy/dx gives an expression for the derivative in terms of both x and y:

dy/dx = (3x²+6x)/(2y)

It should not be surprising that both variables are needed, because the point on the curve is not uniquely determined by x alone. For example, the tangents at the two points with x-coordinate 1 on this cubic have different slopes:

At (1,2), we have dy/dx = (3(1)²+6(1))/(2(2)) = 9/4, and
At (1,-2), we have dy/dx = (3(1)²+6(1))/(2(-2)) = -9/4.

It may seem surprising that only the implicit derivative of the original equation is needed -- if there were a constant in the original equation, it would just disappear. But the value of that constant term helps to determine the value of y, so it is not entirely lost. Indeed, we can consider the graph, in three dimensions, of the equation

z = x³ + 3x² - y²

Picking values for z just amounts to slicing this graph with horizontal planes, forming "contours" of the surface. The plane z=0 gives the present cubic; others, for z=-3,-2,-1,1,2,3, have no crossing points (called nodes, as at (0,0) in the original cubic), though for some the graph is disconnected (when the slicing plane is above the origin but not above the "dome" that has high point at (-2,0,4)):

At any point on any one of these curves, we can use the formula

dy/dx = (3x²+6x)/(2y)

to find the slope of the tangent line. For instance,

the plane z = 3 (outlined in gray) intersects the surface in the disconnected orange curve, including the violet point with coordinates z = 3, x = -2.4, and y equal to the negative square root of .456 (roughly -0.68). The light green tangent line to the orange curve at the violet point in the gray plane has slope given by the formula:

dy/dx = (3(-2.4)²+6(-2.4))/(-2sqrt(.456))

or about -2.1. (The difference in the horizontal and vertical scales in the left picture is a bit misleading. Note that for every horizontal increase of 1 unit, the vertical decrease along the light green line is roughly 2 units.)