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There are quite a few imprecise uses of terminology that could lead to confusion. Addressing these issues may help clarify the various questions raised.

The first imprecise use of terminology appears early in the post:

> Given an implicit function $x^2 + y^3 - 15xy = 0$, ...

While the equation $x^2 + y^3 - 15xy = 0$ can define an implicit function *near* certain points on the curve (as guaranteed by the implicit function theorem), it does NOT define a single function globally.

The figure you provided illustrates the *set* of points in the plane that satisfy the equation: $\{(x, y) : x^2 + y^3 - 15xy = 0\}$. Notably, no implicit function is defined near the point $(0, 0)$ because the equation does not locally represent the graph of a function in terms of either $x$ or $y$.

Another source of confusion is the imprecise phrasing:

> calculate *the* tangent line and *the* vertical tangent line

This phrasing is problematic because:

1. There may be multiple tangent lines, as in the case of a circle, $x^2 + y^2 - 1 = 0$.  
2. It is necessary to specify the tangent line of *which* function and at *which* point.

As noted in the other answer, clarity about the term "tangent line" is crucial. If it refers to the tangent line to the graph of a function at a point, one must explicitly consider neighborhoods where the function is well-defined.

To address these issues, I would revise your question as follows:

> Consider the equation $x^2 + y^3 - 15xy = 0$. Find all horizontal tangent lines to implicitly defined function(s) of $x$ and all vertical tangent lines to implicitly defined function(s) of $y$.

A related issue arises in the following statement:

> Why is using the $\frac{dx}{dy}=0$ method considered better?...

The phrase "$\frac{dx}{dy}=0$ method" is vague and appears to reflect a procedural approach often introduced in introductory calculus courses, where students are somehow encouraged to memorize steps without fully understanding the underlying concepts. However, as discussed earlier, the equation defines multiple implicit functions, so these cases must be considered separately:

- For **horizontal tangent lines**, one should analyze implicit functions of the form $y = f(x)$. Specifically, this involves identifying points where $\frac{dy}{dx} = 0$ using implicit differentiation.

- For **vertical tangent lines**, the focus shifts to implicit functions of the form $x = h(y)$, where one determines points where $\frac{dx}{dy} = 0$ through implicit differentiation.