Clear[f, lag, eqx1, eqx2, eqx3, eqxl, result2, result3, result4, 
      result5, result6]

f[x1_, x2_, x3_] := (x1^((2/5))) (x2^((1/5))) (x3^((2/5)));

lag[x1_, x2_, x3_, l_] :=  
      - {w1, w2, w3} . {x1, x2, x3}  +  
        l (y  -  f[x1, x2, x3])

\(eqx1 = D[lag[x1, x2, x3, l], x1] == 0; 
    eqx2 = D[lag[x1, x2, x3, l], x2] == 0; 

    eqx3 = D[lag[x1, x2, x3, l], x3] == 0; 
    eqxl = D[lag[x1, x2, x3, l], l] == 0; \)

result2 = 
      Solve[{eqx1, eqx2, eqx3, eqxl}, {x1, x2, x3, l}] // Simplify

result3  =  result2[[2]] // Simplify

clong[w1_, w2_, w3_, y_] := 
      -{w1, w2, w3} . {x1, x2, x3} /. result3;

result4 = 
      Map[D[clong[w1, w2, w3, y], #1]&, {w1, w2, w3, y}] // Simplify

This suggests that the derivatives of the cost function with respect to factor prices and products are each factor demand and 
the lagrange multiplier.  The reader should note that the Lagrange multiplier signifies the marginal cost.

Map[D[#1, y]&, {x1, x2, x3, l} /. result3]

result5 = Solve[{eqx1, eqx2, eqxl}, {x1, x2, l}] // Simplify

result6 =   result5[[2]]

cshort[w1_, w2_, w3_, y_] := 
      -{w1, w2, w3} . {x1, x2, x3} /. result6;

cshort[w1, w2, w3, y]