General sensitivity theory is presented for treating problems characterized by systems of nonlinear equations with nonlinear responses. The concept of the Fréchet derivative is shown to be fundamental to both differential and variational approaches. These two approaches, unified through the Fréchet derivative, form an operator viewpoint of sensitivity theory, leading to identical expressions for the adjoint equations and for the sensitivity functions. Also presented is an alternative sensitivity formalism for systems of nonlinear matrix equations, such as those arising from the application of numerical methods to many practical problems. This approach significantly enlarges the scope and versatility of sensitivity theory as it allows direct treatment of parameters that are purely of numerical-methods origin. To demonstrate the usefulness and practical applications of both operator and matrix formalisms, a significantly nonlinear transient problem in fast reactor thermal hydraulics is considered. Following the derivation and comparative analysis of the adjoint equations and sensitivity expressions using both formalisms, an extensive sensitivity study for this problem is presented. Conclusions about the future applicability of the general theory are also discussed.