I was recently studying parabolas $ f(x) = ax^2 + bx + c $ whose derivative $f'(x) = 2ax + b$ is tangent to itself -- one example would be $f(x) = x^2 -6x +10;$ it is easy to see that if $c = a + \frac{b^2}{4a},$ then the parabola will hold this property, and $f'(x)$ is tangent at $\left( 1-\frac{b}{2a}, 2a\right).$
I wanted to ask if anyone knew of other real functions aside from parabolas (or even polynomials) whose derivative is tangent to itself. (This is excluding trivial cases such as $f(x) = 0$ or $f(x) = Ce^x$).
Follow-up question: If such other functions exist, does there exist one in which the derivative is tangent at a countably infinite number of points (e.g., in the manner that the line $y=1$ is tangent to $\sin(x)$ or $\cos(x)$)?
Thank you kindly!