It was Joseph-Louis Lagrange (1736-1813) who defined the *derived function* or derivative \( f'(x) \) of a function \( f(x) \). Derivatives can be defined in terms of limits, as we do today. Furthermore, derivatives and differentials are to a great extent interchangeable when talking about functions \( y=f(x) \). On one hand, if you know \( dy \) in terms of \( dx \), then you can solve for \( f'(x) \) by dividing the expression for \( dy \) by \( dx \). On the other hand, if you know how to find \( f'(x) \), then \( dy=f'(x)dx \), a familiar relation from integration by substitution.

**Figure 3.** Joseph-Louis Lagrange (print in public domain).

However, Lagrange promoted a third path towards the foundations of calculus involving power series expansions. If \( f(x) \) has a Taylor series expansion at \( x=x_0\), then the coefficient of the increment \( x - x_0 \) in the series expansion is \( f'(x_0)\). Lagrange used this property to *define* the derivative. This approach seems hopelessly circular to modern readers, because we think of power series expansions as given by Taylor's theorem, and therefore presupposing not only the first derivative \( f'(x) \) but also all the higher derivatives. However, through the course of the 17th and 18th centuries, mathematicians were able to give power series expressions for all of the familiar functions of calculus without the explicit use of derivatives. As an example, the geometric series

\[ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \ldots \]

can be derived by long division, or by considering the telescoping expansion of \( (1-x)(1 + x + x^2 + \ldots + x^{n-1}) \).

For more examples, see [Euler 1748], where Leonhard Euler (1707-1783) demonstrates various tricks for deriving the power series for trigonometric, exponential, and logarithmic functions. If an arbitrary function \( f(x) \) has a power series expansion at every point \( x_0 \) in its domain, Lagrange could define \( f'(x) \) without recourse to either infinitely small increments or limits. In his *Theory of Analytic Functions* [Lagrange 1797], he gave a general method for deriving the power series of a large class of analytic functions. For more on Lagrange's foundational program, see Victor Katz's *History of Mathematics* [2009, pp. 633-636].