How does tune ν and centroid orbit Δx change with k2, when corrector coils can be changed with k2 changes :

(xn+1+Δx,xn+1'+Δx') = M.((xn+Δx,xn'+Δx')+(c(t),c'(t))+(({{0}, {(1/2)k2(t) (((xn+Δx))^2)}})))

(Δx,Δx') = M.((Δx,Δx')+(c(t),c'(t))+(({{0}, {(1/2)k2(t) (((Δx)^2)+(σ^2))}})))

(xn+1,xn+1') = M.((xn,xn')+(({{0}, {k2(t) xn Δx}})))

Tune of the beam :

Δν=1/2β k2(t) Δx

Centroid of the beam without the sextupole would be co(t):

(Δx,Δx') = M.((Δx,Δx')+(c(t),c'(t))+(({{0}, {(1/2)k2(t) (((Δx)^2)+(σ^2))}})))

(Δx,Δx')=((((M^(-1))-1))^(-1))(({{c(t)}, {(1/2)k2(t) (((Δx)^2)+(σ^2))+c'(t)}}))

(Δx,xxx)=(co(t),co'(t))+(1/2)(({{xxx, β Cot[(μ/2)]}, {xxx, xxx}}))(({{0}, {(1/2)k2(t) (((Δx)^2)+(σ^2))}}))

Δx=co(t)+((k2(t))/4)β Cot[(μ/2)](((Δx)^2)+(σ^2))

(1) Strategy without tune change, measure orbit compensation.
Δx: centroid of the beam.
co: centroid if there were no sextupole, is being changed with k2.
k2: sextupole strength being changed.

Δx(co, k2,σ)
Δν(k2,Δx) = Δν(co,k2,σ)

Change co with k2 so that Δν=0, leading to co(k2,σ).
Change k2, measure co(k2,σ) by means of the corrector coils creating the co and fit σ to the measurements.

Δx=co(t)+((k2(t))/4)β Cot[(μ/2)](((Δx)^2)+(σ^2))
Δx = (2/(β k2))Tan[(μ/2)] - (Sqrt[((((2/(β k2))Tan[(μ/2)]))^2)-(4/(β k2))Tan[(μ/2)]co-(σ^2)])
Δν=β k2(t) Δx=2Tan[(μ/2)] - (Sqrt[(((2Tan[(μ/2)]))^2)-β k2 4Tan[(μ/2)]co-(((β k2))^2)(σ^2)])
co=-((β k2)/4)Cot[(μ/2)](σ^2)
(σ^2) =|(dco/dk2)Tan[(μ/2)](4/β)|

(2) Futile Strategy without tune change, measure orbit change.
Δx(co, k2,σ)
Δν(k2,Δx) = Δν(co,k2,σ)

Change co with k2 so that Δν=0, leading to co(k2,σ) and to the orbit Δx(co(k2,σ), k2,σ)
Change k2, measure Δx(k2,σ) and fit σ to the measurements.
co=-((β k2)/4)Cot[(μ/2)](σ^2)
Δx=co(t)+((k2(t))/4)β Cot[(μ/2)](((Δx)^2)+(σ^2))=((k2(t))/4)β Cot[(μ/2)]((Δx)^2)
Δx = 0 from which σ cannot be fit.

(3) Strategy with tune change while k2 changes from 0.

Δx(co, k2,σ)
Δν(k2,Δx) = Δν(co,k2,σ)

Do not change co with k2.
δx = Δx - co, the closed orbit change from exciting the sextupole.
Measrue δx(co, k2,σ) and Δν(co,k2,σ) and fit to co and σ.

δx=((k2(t))/4)β Cot[(μ/2)]((((δx+co))^2)+(σ^2))
δx = (2/(β k2))Tan[(μ/2)] - (Sqrt[((((2/(β k2))Tan[(μ/2)]))^2)-(4/(β k2))Tan[(μ/2)]co-(σ^2)]) - co
Δν=β k2 Δx=2Tan[(μ/2)] - (Sqrt[(((2Tan[(μ/2)]))^2)-β k2 4Tan[(μ/2)]co-(((β k2))^2)(σ^2)])
co = ((Δν)/(β k2))-δx
(σ^2) = δx (4/(β k2))Tan[(μ/2)]-(((((Δν)/(β k2))))^2)

(4) Strategy with tune change while k2 changes from its lattice setting.

δx(co, δk2,σ) = Δx(co, k2+δk2,σ) - Δx(co, k2,σ)
δν(co,δk2,σ) = Δν(co,k2+δk2,σ) - Δν(co,k2,σ)

Do not change co with δk2.
Measrue δx(co, δk2,σ) and δν(co,kδ2,σ) and fit to co and σ.

Δx+δx=co+((k2+δk2)/4)β Cot[(μ/2)]((((Δx+δx))^2)+(σ^2))
Δx=co+(k2/4)β Cot[(μ/2)](((Δx)^2)+(σ^2))
δx = ((δk2)/4)β Cot[(μ/2)]((((Δx+δx))^2)+(σ^2))+(k2/4)β Cot[(μ/2)](((δx)^2)+2Δx δx)
δx = (1/(β (k2+δk2)))(2Tan[(μ/2)] - (Sqrt[(((2Tan[(μ/2)]))^2)-4 β (k2+δk2)Tan[(μ/2)]co- ((β )^2)(((k2+δk2))^2)(σ^2)])-(1/(β k2))(2Tan[(μ/2)] + (Sqrt[(((2Tan[(μ/2)]))^2)-4 β k2Tan[(μ/2)]co-(((β k2))^2)(σ^2)])
δν=( - (Sqrt[(((2Tan[(μ/2)]))^2)-4 β (k2+δk2)Tan[(μ/2)]co-(β^2) (((k2+δk2))^2)(σ^2)])) + (Sqrt[(((2Tan[(μ/2)]))^2)-4 β k2Tan[(μ/2)]co-(((β k2))^2)(σ^2)])) )

δx β (k2+δk2) - δν =((δk2)/k2)( -2Tan[(μ/2)] +(Sqrt[(((2Tan[(μ/2)]))^2)-4 β k2Tan[(μ/2)]co-(((β k2))^2)(σ^2)]))
δx β k2 - δν = (-2Tan[(μ/2)] + (Sqrt[(((2Tan[(μ/2)]))^2)-4 β (k2+δk2)Tan[(μ/2)]co- (((k2+δk2))^2)(σ^2)]))(1-(1/(1+δk2/k2))) 

((δν - δx β (k2+δk2) )/(δk2)) (k2/2) Cot[(μ/2)] =  1 - (Sqrt[1-β k2 Cot[(μ/2)]co-(((β (k2/2) Cot[(μ/2)]))^2)(σ^2)]) = A 
((δν - δx β k2 )/(δk2)) (k2/2) (1+((δk2)/k2))Cot[(μ/2)]=  1 - (Sqrt[1-β (k2+δk2) Cot[(μ/2)]co-(((β (((k2+δk2))/2) Cot[(μ/2)]))^2)(σ^2)]) = B

(((A-1))^2)-1+β k2 Cot[(μ/2)]co+(((β (k2/2) Cot[(μ/2)]))^2)(σ^2) = 0
(((B-1))^2)-1+β (k2+δk2) Cot[(μ/2)]co+(((β (((k2+δk2))/2) Cot[(μ/2)]))^2)(σ^2) = 0
(((A - 1))^2)(1+((δk2)/k2)) - (((B-1))^2)- ((δk2)/k2)+ (((β (k2/2) Cot[(μ/2)]))^2)(σ^2)(1+((δk2)/k2))(-((δk2)/k2))=0

(((((δν - δx β (k2+δk2) )/(δk2)) (k2/2) Cot[(μ/2)]-1))^2)(1+((δk2)/k2))-(((((δν - δx β k2 )/(δk2)) (k2/2) Cot[(μ/2)](1+((δk2)/k2))-1))^2)-((δk2)/k2)=(((β (k2/2) Cot[(μ/2)]))^2)(σ^2)(1+((δk2)/k2))(((δk2)/k2))

(((((δν - δx β k2 )/(δk2)) (k2/2) Cot[(μ/2)]-1-δx β  (k2/2) Cot[(μ/2)]))^2)(1+((δk2)/k2))-(((((δν - δx β k2 )/(δk2)) (k2/2) Cot[(μ/2)]-1+(δν - δx β k2 ) (1/2) Cot[(μ/2)]))^2)-((δk2)/k2)=(((β (k2/2) Cot[(μ/2)]))^2)(σ^2)(1+((δk2)/k2))(((δk2)/k2))

(((((δν - δx β k2 )/(δk2)) k2 -2Tan[(μ/2)]-δx β  k2 ))^2)(1+((δk2)/k2))-(((δν+((δν - δx β k2 )/(δk2)) k2 -2Tan[(μ/2)] - δx β k2))^2)-4((δk2)/k2)((Tan[(μ/2)])^2)=(((β k2 ))^2)(σ^2)(1+((δk2)/k2))(((δk2)/k2))

(σ^2)=(((((δx k2 )/(δk2))))^2)(1+((δk2)/k2))+((δx)/(δk2))(1/β)Tan[(μ/2)]-(((((δν/β)/(δk2))))^2)

For k2 = 0:

(σ^2)=((δx)/(δk2))(4/β)Tan[(μ/2)]-(((((δν/β)/(δk2))))^2)

To leading order for k2 not  0:

(σ^2)=((δx)/(δk2))(4/β)Tan[(μ/2)]-(((((δν/β)/(δk2))))^2)+(((((δx k2 )/(δk2))))^2)(1+((δk2)/k2))

Since the final formula is simple, there should be a simpler derivation.

δν=β(k2+δk2)(Δx+δx)-βk2 Δx=β (k2+δk2) δx+β δk2 Δx
((δν-k2 β δx(1+((δk2)/k2)) )/(δk2 β))=Δx
((δν-k2 β δx )/(δk2 β))=Δx+δx

Δx+δx=co+((k2+δk2)/4)β Cot[(μ/2)]((((Δx+δx))^2)+(σ^2))
Δx=co+(k2/4)β Cot[(μ/2)](((Δx)^2)+(σ^2))
δx=((k2+δk2)/4)β Cot[(μ/2)]((((Δx+δx))^2)+(σ^2))-(k2/4)β Cot[(μ/2)](((Δx)^2)+(σ^2))
(σ^2)=(4/β)((δx)/(δk2)) Tan[(μ/2)]-((((Δx+δx))^2)(1+((δk2)/k2))-((Δx)^2))(k2/(δk2))
(σ^2)=(4/β)((δx)/(δk2)) Tan[(μ/2)]-((((((δν-k2 β δx )/(δk2 β))))^2)(1+((δk2)/k2))-(((((δν-k2 β δx(1+((δk2)/k2)) )/(δk2 β))))^2))(k2/(δk2))

(σ^2)=(4/β)((δx)/(δk2)) Tan[(μ/2)]-(((((δν )/(δk2 β))))^2)+(((((k2 δx)/(δk2))))^2)(1+((δk2)/k2))