Explicit Function

Some implicit relations convert. The circle $x^2 + y^2 = 25$ splits into $y = \sqrt{25 - x^2}$ for the top half and $y = -\sqrt{25 - x^2}$ for the bottom half, because one equation can hold more than one function inside it. Neither branch is the circle; each is a piece of it.

That one does not convert. No rearrangement in terms of the usual functions isolates $y$, yet $y$ still varies with $x$ and is differentiable except where $\cos y = -1$. Differentiating both sides gives $\frac{dy}{dx} + \cos y \cdot \frac{dy}{dx} = 1$, so $\frac{dy}{dx} = \frac{1}{1 + \cos y}$. That formula fails at the isolated inputs $x = \pi + 2k\pi$, where $y = \pi + 2k\pi$ makes $1 + \cos y = 0$ and the curve has a vertical tangent. Between those points the derivative is in hand while an explicit formula is not, which is the whole reason implicit differentiation exists.

Implicit differentiation is not the only door, only the one that is always open. Here $x$ is already isolated, so $\frac{dx}{dy} = 1 + \cos y$ and the inverse function rule returns the same $\frac{dy}{dx} = \frac{1}{1 + \cos y}$ with no implicit step. The circle yields to a parametrisation instead: $x = 5\cos t$, $y = 5\sin t$ gives $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = -\cot t$ without ever solving for $y$.