If a Lagrangian depends on second and higher order derivatives of the fields, \(L=\) \(L\left(\Phi_{\lambda}, \Phi_{A \mu}, \Phi_{A, \mu \nu}, \ldots\right)\) derive the generalized Euler-Lagrange equations $$ \frac{\delta L \sqrt{-g}}{\delta \Phi_{A}} \equiv \frac{\partial L \sqrt{-g}}{\partial \Phi_{A}}-\frac{\partial}{\partial x^{\mu}}\left(\frac{\partial L \sqrt{-g}}{\partial \Phi_{A, \mu}}\right)+\frac{\partial^{2}}{\partial x^{\mu} \partial x^{\prime}}\left(\frac{\partial L \sqrt{-g}}{\partial \Phi_{A_{L} w}}\right)-\cdots=0 $$

The Euler-Lagrange equations are fundamental equations that are extremely useful when dealing with problems in the calculus of variations, and are given by the formula:\[\frac{\partial L}{\partial \Phi_{A}} - \frac{\partial}{\partial x^{\mu}}\left(\frac{\partial L}{\partial \Phi_{A, \mu}}\right)=0\]However, in this problem, the Lagrangian \(L\) depends also on the second and higher order derivatives of \(\Phi_{A}\), the field. Therefore, additional terms appear in the Euler-Lagrange equations, one for each additional derivative of the field that appears in the Lagrangian.

Applying the Euler-Lagrange equations to the given Lagrangian:\[\frac{\partial L \sqrt{-g}}{\partial \Phi_{A}} - \frac{\partial}{\partial x^{\mu}} \left(\frac{\partial L \sqrt{-g}}{\partial \Phi_{A, \mu}}\right) + \frac{\partial^{2}}{\partial x^{\mu} \partial x^{\nu}} \left(\frac{\partial L \sqrt{-g}}{\partial \Phi_{A, \mu \nu}}\right) - \cdots = 0\]Note that for each additional derivative of the field in the Lagrangian, there is an additional derivative on the partial derivatives in the Euler-Lagrange equation. Likewise, for each subsequent derivative, there will be a corresponding term in the Euler-Lagrange equations. Additionally, because the action should be invariant under local changes on the metric, the addition of a term \(-g\), the determinant of the metric tensor, accompanied by a square root, ensures that this feature is preserved.

Therefore, using the given Euler-Lagrange equations:\[\frac{\partial L \sqrt{-g}}{\partial \Phi_{A}} - \frac{\partial}{\partial x^{\mu}}\left(\frac{\partial L \sqrt{-g}}{\partial \Phi_{A, \mu}}\right) + \frac{\partial^{2}}{\partial x^{\mu} \partial x^{\nu}}\left(\frac{\partial L \sqrt{-g}}{\partial \Phi_{A, \mu \nu}}\right) - \cdots = 0\]The given Lagrangian \(L\), which depends on second and higher order derivatives of the fields, will satisfy this equation when it is minimized.