История изменений

n = 3;
X = Array[x, n];
Y = Array[y, n];
Z = Array[z, n];
Solve[D[Sum[(y[t] - (x[t]*k + z[t]*(1 - k)))^2, {t, n}], k] == 0]

res:
{{y[3] -> (1/(
   x[3] - z[3]))(k x[1]^2 + k x[2]^2 + k x[3]^2 - x[1] y[1] - 
     x[2] y[2] + x[1] z[1] - 2 k x[1] z[1] + y[1] z[1] - z[1]^2 + 
     k z[1]^2 + x[2] z[2] - 2 k x[2] z[2] + y[2] z[2] - z[2]^2 + 
     k z[2]^2 + x[3] z[3] - 2 k x[3] z[3] - z[3]^2 + k z[3]^2)}, {x[
    3] -> z[3], 
  y[2] -> (1/(
   x[2] - z[2]))(k x[1]^2 + k x[2]^2 - x[1] y[1] + x[1] z[1] - 
     2 k x[1] z[1] + y[1] z[1] - z[1]^2 + k z[1]^2 + x[2] z[2] - 
     2 k x[2] z[2] - z[2]^2 + k z[2]^2)}, {x[1] -> z[1], x[2] -> z[2],
   x[3] -> z[3]}, {x[2] -> z[2], x[3] -> z[3], 
  y[1] -> k x[1] + z[1] - k z[1]}}

n = 3;
X = Array[x, n];
Y = Array[y, n];
Z = Array[z, n];
D[Sum[(y[t] - (x[t]*k + z[t]*(1 - k)))^2, {t, n}], k]

res:
2 (-x[1] + z[1]) (-k x[1] + y[1] - (1 - k) z[1]) + 
 2 (-x[2] + z[2]) (-k x[2] + y[2] - (1 - k) z[2]) + 
 2 (-x[3] + z[3]) (-k x[3] + y[3] - (1 - k) z[3])