Sunday, 8 October 2017

differentiation - Derivative of the product of operators and Derivative of exponential




  1. I'm asked to show that d(ˆAˆB)dλ = dˆAdλˆB+ˆAdˆbdλ

    With λ a continuous parameter. Should I use the definition dˆAdλ = limϵ0ˆA(λ+ϵ)ˆA(λ)ϵ
    applied to ˆAˆB like d(ˆAˆB)dλ = limϵ0ˆA(λ+ϵ)ˆB(λ+ϵ)ˆA(λ)ˆB(λ)ϵ
    and do some algebra to get the RHS of the first equation, or I'm missing something?





  2. Another interesting derivative to pay attention to is: ddλexp(ˆA(λ)) ?





Answer



A(λ+ϵ)B(λ+ϵ)=(A(λ)+ϵ˙A)(B(λ)+ϵ˙B)=A(λ)B(λ)+ϵ(˙AB+A˙B)+o(ϵ2)


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