Reading old math papers and refreshing my geometry has lead me to the following:
Question: What is the largest class of curves $f:[a,b]\to \mathbb{R}$ whose tangent line at a point $x_0\in [a,b]$ can be computed solely using analytic geometry and algebra? And what methods or theory might we use to prove the impossibility for curves outside of this class (analogous to the impossibility of famously desired Euclidean constructions)?
Some examples: For example, if $f(x)=x^2$ is a parabola, then we can find the points of intersection with a line $y=mx+b$, obtaining that they must be roots of the polynomial $p(x)=x^2-mx-b$ and for this root to be unique, the discriminant must be zero, i.e. $m^2+4b=0$ or $b=-m^2/4$ and thus the unique point of intersection is at $x=m/2$. Thus given the point $x_0$, the slope of the tangent line at $x_0$ is $m=2x_0$ and the equation of the tangent line at the point $x_0$ is then $$y=2x_0 x-x_0^2.$$
To find the slope of the tangent line of higher powers like $x^5$ would require solving $$x^5-mx-b=0,$$ which I do not believe has closed form solutions. So already, we are pretty limited.
We can also find the tangent to a circle in this way. Indeed, if $x^2+y^2=r^2$ and $y=mx+b$, the points of intersection are roots of the polynomial $$p(x)=x^2+\frac{2mb}{1+m^2} x+\frac{b^2-r^2}{1+m^2}.$$ The same steps are applied, set the discriminant to zero, solve for $b$ and then solve for $m$ from just the first term of the quadratic equation. This gives that $b=r\sqrt{1+m^2}$ and that $$x=-\frac{rm}{\sqrt{1+m^2}},$$ which when we invert, being judicious of signs, gives the slope of the tangent line at $x$ as $$m=\frac{-x}{\sqrt{r^2-x^2}},$$ for $0<x<r$.
So the class $C$ of such curves at least contains curves $f$ such that $p(x)=f(x)-mx-b$ is a second-degree polynomial and that $b$ can be chosen such that the discriminant of $p$ is zero, resulting in an equation $x=g(m)$ that can then potentially be inverted to solve for $m$.
Are there others? Is this exhaustive? This idea is interesting to me, for both historical purposes, and possibly pedagogical purposes, since just as the impossibility of certain geometric constructions shows the limitations of straightedge and compass Euclidean geometry, these ideas might elude to the limitations of analytic geometry and algebra without the notion of limits, fluxions, or infinitesimals. Of course, this is nothing new or groundbreaking, but is fun to think about and review.