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 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,* i.e., *y =* an
expression in terms of *x* only). 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
*y ^{2}=x^{3}+3x^{2}* are functions
of

In computing the derivative of the left side, when the Chain
Rule demanded that we write the derivative of
*y* with respect to *x,* we could only write *dy/dx*,
since we do not know *y* explicitly as a function of *x*.
But we now have an equation in which the very quantity we
wanted, *dy/dx* appears, and in fact it is easy to solve
the equation for it -- though the equation may be otherwise
complicated, it is linear in *dy/dx.* Solving for
*dy/dx* gives an expression for the
derivative, in terms of both *x* and *y:*

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

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

to find the slope of the tangent line. It amounts to
finding the slope of a horizontal line tangent to the
surface at some point -- the slope being computed in the
horizontal *xy*-plane that cuts the surface at
that point.