We present a differential-calculus-based method which allows one to derive more identities from a given Fibonacci-Lucas identity containing a finite number of terms and having at least one free index. The method has two independent components. The first component allows new identities to be obtained directly from an existing identity while the second yields a generalization of the existing identity. The strength of the first component is that no additional information is required about the given original identity. We illustrate the method by providing new generalizations of some well-known identities such as d'Ocagne's identity, Candido's identity, the Gelin-Cesàro identity, and Catalan's identity. The method readily extends to a generalized Fibonacci sequence.