When you go with x x[t]r-r sin[t] y y[t] = r-rcos [t], as in the basics, you get a cycloid. calculating y in the form y = f[x] is almost out of the question nevertheless, it's not a big deal to calculate f'[x] when you are given a specific value of x here is a table in the form fx, f' [x]} for selected x 's. in 7 clear[r, t, f] table[{n[r t - r sin[t]], f'[n[r t - r sin[t]]]}, {t, 0, 2 pi, pi/6}] out7= {{0., f[0.] }, {0.0235988 r, f' [0.0235988 r]}, {0.181172 r, f' [0.181172 r] }, (0.570796 r, f' [0.570796 r]}, {1.22837 r, f'[1.22837 r] }, 2.11799 r, f' [2.11799 r]]}, {3.14159 r, f'[3.14159 r] }, {4.16519 r, f' [4.16519 r] }, {5.05482 r, f'[5.05482 r] }, (5.71239 r, £' [5.71239 r]}, {6.10201 r, £' [6.10201 r] }, {6.25959 r, f' [6.25959 r]}, {6.28319 r, f' [6.28319 r] }} unfortunately f'[x] has not been calculated simply because no clean formula for f; x] is available. copy, paste, and edit the last instruction so that each second slot exhibits the actual value of f'[xl for the corresponding x in the first slot