Calculation of second derivative of Rayleigh quotient

Hello everyone. I have an eigenvalue problem of the form (A-kB)V=0 and I calculate the eigenvalues k using the Rayleigh quotient such as k=trans(Vl)*A*Vr/trans(Vl)*B*Vr, where Vl and Vr are the left and right eigenvectors respectively. The matrices A and B are functions of a variable x. What I want is to calculate the first and second derivative of k with respect to x, using the expression given by the Rayleigh quotient. So far, I have completed the calculation of the first derivative, which is given by the expression dk/dx=((Vl'*A,x*Vr)*(Vl'*B*Vr)-(Vl'*A*Vr)*(Vl'*B,x*Vr) )/(Vl'*B*Vr)^2 and this gives me correct results, compared to those extracted using central differences. However, I face a difficulty calculating the second derivative, since it gives me different results from those computed by central differences. Has anyone done this before or has any similar experience? Any advice could be very useful. It is worth mentioning that my A and B matrices have trigonometrical terms of cos(x), sin(x) and products of them.

Thank you in advance!

[Antigoni]

Is there any limit is using the Rayleigh quotient or it's always applicable?