Suppose we are given a function $f(x,y) = x^2 + y$ defined over $\mathbb{R}^2$. Then $$\frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \cdot \frac{dy}{dx} = 2x + \frac{dy}{dx}$$ and $$\frac{\partial f}{\partial x} = 2x$$

**How do I interpret $\frac{df}{dx}$, both mathematically and intuitively?**

We know derivatives refer to tangents and in the case of $\frac{\partial f}{\partial x}$, we fix the $y$ and look at the tangent of the now-single-variable function $g(x) = f(x,y\_0)$. I don't know what it means in the case of $\frac{df}{dx}$.