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Show Osculating Circle

t = 1.5707963267948966 (slider from 0 to 31.41592654, step 0.01)
f(x) = x * sin(x) + x / 2
k: sin(t)^(3)
A = Point(f)
a = Tangent(A, f)
c = OsculatingCircle(A, f)
f'(x) = f'(x)
f''(x) = f''(x)
b: 1 / Curvature[A, f]
B = Point(f)
h: x(B) - x(A)
r: sqrt(1 + f'(x(A))^(2)) * (h^(2) + (f(x(A)) - f(x(A) + h))^(2)) / (2 * f(x(A) + h) - 2 * f(x(A)) - 2 * h * f'(x(A)))
t_1: r / sqrt(1 + f'(x(A))^(2))
C: (x(A) - t_1 * f'(x(A)), y(A) + t_1)
e = OrthogonalLine(A, a)
j = true
d = If(h ≠ 0, Circle[C, abs(r)], OsculatingCircle[A, f])
n = 1