Rates of change and derivatives
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Ever since Lagrange initiated a new epoch in pure analysis by defining the derivative of a function, the logical clarity of applied mathematics has suffered from a confusion of those derivatives with the rate of change of one variable quantity with respect to another. Yet a mere count of the ideas involved in the two concepts clearly demonstrates that the situations studied in pure and in applied mathematics are basically unlike. The derivative associates a function with one function; for instance, the cosine function with the sine function. The rate of change associates a variable quantity with two variable quantities; for instance, the velocity with the distance travelled and the time.
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