Numerically for the curve given any kind of equations. With geom3d  you construct the plane equation for any three points of the curve that does not lie on one line. In a cycle of successively take point of the curve, substitute them in the equation of the plane and monitor the absolute value of the discrepancy.

Many choices on the same basis.
PAR.mw

restart:
a := [x+1, x+2, x+3, x+4];
a := convert(a, set);
a :=minus(a, {x+2});
a := convert(a, list);

restart;
a := [x+1, x+2, x+3, x+4];
a := subs(x+2 = NULL, a);
nops(a);
op(2, a);

restart: with(plots):
t := [1, 2, 3];
f(t[1]) := [3, 4]; f(t[2]) := [11, 12]; f(t[3]) := [41, 1];
pointplot([f(t[1]), f(t[2]), f(t[3])], color = RGB(7, .3, 4), style = line, symbol = solidcircle, thickness = 5);
pointplot3d([1, op(f(t[1])), 2, op(f(t[2])), 3, op(f(t[3]))], color = RGB(7, .3, 4), style = line, symbol = solidcircle, thickness = 5);

combine(-ln(x)+ln(y), symbolic);

restart;
f := x1^2/(x2^3*x3^2);
op(1, op(2, f))^sign(op(2, op(2, f)));
op(1, op(3, f))^sign(op(2, op(3, f)));
f := algsubs(1/x2 = x2b, f);
f := algsubs(1/x3 = x3b, f);

restart;

(diff(f(x), x))/(diff(ln(x), x));

for example:

restart;

f := sin(x);

(diff(f, x))/(diff(ln(x), x));

Skeptik18(_for_d1.mw

 For a start point “a” = 0.5 any number of solutions. It depends on the "smax".

   1, [(0.8013209420000008)], 2.70183810879842667*10^-8                                                            

   2, [(1.0938038328000006)], 1.75716205141895898*10^-7                                                        

      3, [(1.5165511908)], 3.19181676505797540*10^-8                                                        

   4, [(1.9061358998000002)], 4.58627091859398206*10^-9                                                           

   5, [(2.1833650214000007)], 4.36816931514982798*10^-8                                                            

   6, [(2.5877177300000005)], 1.71876049503971729*10^-7                                                            

    7, [(2.937538889999998)], 9.29630750157173224*10^-9                                                           

    8, [(3.219756611999996)], 2.01128799837135830*10^-7                                                            

   9, [(3.6180527559999964)], 4.55144026911824540*10^-8                                                           

   10, [(3.953152181999997)], 3.39347172584325562*10^-8                                                            

   11, [(4.239272189999994)], 2.57192325658905930*10^-7                                                           

   12, [(4.635025125999992)], 3.01111789280383846*10^-7                                                         

   13, [(4.962516473999992)], 9.20600748965938465 10^-8                                                            

   14, [(5.2514410039999975)], 2.19850920579744980*10^-7                                                           

   15, [(5.645899321999999)], 1.82685308214303177*10^-7                                                         

   16, [5.968760335999992)], 2.30153605063065925*10^-8                                                          

   17, [6.2597571959999945)], 2.33606770816408017*10^-7                                                           

   18, [(6.653469225999989)], 1.16942534766906194*10^-7                                                           

    19, [(6.97322109999999)], 7.71696275769784279*10^-8                                                          

   20, [(7.265801703999997)], 4.86139439814792240*10^-8                                                           

   21, [(7.659044969999998)], 4.08131656026711200*10^-7                                                          

   22, [(7.976567196000005)], 6.48611420128730742*10^-8                                                    

   23, [(8.270394148000008)], 1.66075897922723926*10^-7                                                           

   24, [(8.663323925999999)], 2.60858493916771295*10^-7                                                      

      25, [(8.979170028)], 7.82206077687419565*10^-8                                                           

   26, [(9.274002000000005)], 1.76298078358172460*10^-8                                                           

   27, [(9.666711878000001)], 5.96099682503847818*10^-7                                                        

      28,[(9.981252484)], 2.84982932807764656*10^-8                                                            

   29, [(10.276911416000011)], 2.49167541710448860*10^-7 

...

The Draghilev method. Read, for example: http://www.mapleprimes.com/posts/145360-The-Dragilev-Method-1-Some-Mathematical

restart;
Digits := 30; 2^29.403243784;

For example:

restart:
nn := nextprime(10^100); zz := 1;
for ii from 0 to 100000 do zz := `mod`(zz^2+1, nn); if `or`(ii > 99997, zz = 66388502) then print("ii=", ii, "zz=", zz) end if end do:

Blue - the denominator sin(x+(1/3)*Pi-theta) = 0.
theta, I think, has a period of Pi, and x has a period of Pi / 3. The solution is obtained by Draghilev method. This numerical solution of ordinary differential equations with initial conditions theta (0) = 0, x (0) = Pi / 3.

METHOD(n-1)2d.mw

MaplePrime.mw

This way you get all the real solutions for any real value of any parameter for the polynomial

equations N * N.

The Draghilev method to find all the points of zero and Pi/2 slope of an implicit function f(x1,x2)=0 while driving along the section of the line connected.

Example solution of equation and searching of all the points of zero and Pi/2 slope (with animation):

x1^3+x2^3-0.1e-1*sin(1.00001*x1+x2)=0;

D__LIST0.mw

The Draghilev method for F(x,y,z)=0. Many points with good precision (variants with h and h1).

diffanimationAA.mw