A visual proof for a function whose derivative in any point is equal to the distance from the origin, pretty sure I'm the first to discover it, i.e. $$f(\sqrt{2x})=\sqrt{\frac x2}\frac{J_{5/4}(x)+J_{3/4}(x)-J_{-5/4}(x)-J_{-3/4}(x)}{J_{1/4}(x)+J_{-1/4}(x)}-\frac 1{\sqrt{8x}}$$