Problem: Find the tangent line at the point (0,2) to the curve given by the implicit equation

x⁴ − 4xy + y ⁴ = 16.

Solution: We can prove using the Implicit function theorem (see Functions - Theory - Implicit and parametric functions - Implicit functions) that on a neighborhood of the given point (0,2) the given equation in fact defines a function. The equation of the tangent line needs knowledge of a point - which we have - and the slope, which is given by the derivative of the function defining the graph.

However, it is not possible to express y as a function of x explicitly from the given equation. Thus we find its derivative using implicit differentiation. First we take the given equation and differentiate both sides with respect to x, keeping in mind that y is in fact a function of x.

4x³ − 4y − 4xy′ + 4y ³y′ = 0.

Now we solve this equation for y′, then substitute into it x = 0 and y = 2.

Now we know the slope of the tangent line and we can write its equation:

y − 2 = (1/4)⋅(x − 0),     hence     y = (1/4)⋅x + 2.

Remark: Finding out the shape of implicit curves is rather tricky. Probably the only thing one can see reasonably easily from the given equation is that the expression on the left is symmetric with respect to x and y, which means that the resulting shape will be symmetric with respect to the origin.

Using advanced tools (or a suitable software) it turns out that the shape looks like a rounded convex diamond elongated along the main diagonal.

By the way, the elongation is given by the middle term on the left in the equation. Imagine that we put "Axy" instead of "− 4xy" in that equation. How does changing A influence the picture? The sign of A determines the orientation. For A negative the shape gets stretched along the main diagonal (as in our example), for A positive it gets stretched along the other diagonal. The size of A (in absolute value) determines the extent of that elongation, for A = 0 we thus get a shape that is not distorted in any direction, it is a rounded convex square.

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