There is, of course, nothing new about Cauchy's calculations of derivatives. Nor is there anything particularly new about the theorems Cauchy was able to prove about derivatives. Lagrange had derived the same results from his own definition of the derivative. But because Lagrange's definition of a derivative rested on the false assumption that any function could be expanded into a power series, the significance of Cauchy's works lies in his explicit use of the modem definition of a derivative, translated into the language of inequalities through his definition of limit, to prove theorems. The most important of these results, in terms of its later use, was in Lesson 7.
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