Is there a way to show this other than with calculus and implicit differentiation? I am able to easily do this, but I was wondering if there was another way to show it.
I feel I may be missing something obvious or going down unnecessary rabbit trails because I have tried using the Diophantus chord method.
Using the chord method, I have that the equation of the tangent line would have the form $Y=t(X-3)+5$ where t is some rational slope.
I then substituted this into $0=X^3-Y^2-2$.
Eventually, I get to $0=(X-3)(X^2+(3-t^2)X+(9+3t^2))$.
The first factor gives the known solution of $X=3$ and I tried to use the quadratic formula to solve the second factor but then only led me a nasty looking expression with a square root, which leads me to believe I am wrong since I am looking only for rational solutions. I also feel that I have gone down the wrong path because even if I do have success with the chord method, I would only have parameterized the rational solutions to $y^2=x^3-2$ and not actually found the equation of the tangent line.
If someone could guide me toward a method of finding the tangent to $y^2=x^3-2$ at (3,5) without the use of calculus (or point out what I am missing) that would be awesome. Thank you!