The Monod growth function is used extensively in models from the life sciences. It was proposed by Jacques Monod to describe the rate of microbial growth as a function of nutrient density (see Example 2.2.1 of the textbook and its associated ‘Biology Background’ material on this website). Example 3.4.5 in the textbook calculates the derivative of this function. The animation below allows you to explore the graph of the function (in black) along with the graph of its derivative (in red) for different values of the constants.
Notice that the derivative is alway positive and it approaches zero as N gets large, regardless of the values of the constants. If we view R as a model of microbial growth as Monod did, then these properties of the derivative reflect the fact that adding more nutrient increases growth (the derivative is positive) but that there are diminishing returns to adding more and more nutrient (the derivative decreases in magnitude and approaches zero). Put another way, the effect of adding more nutrient on growth is small when the nutrient level is already high.
You can also see from the formula for dR/dN calculated in Example 3.4.5 of the textbook that the magnitude of the derivative (the effect of adding more nutrient) is larger when the constant S is larger. Verify this using the above animation and think about what this means biologically (recall from Example 2.2.1 of the textbook and the associated ‘Biology Background’ material on this website that S can be interpreted as the maximum possible growth). You can also verify that the magnitude of the derivative dR/dN changes in a more complicated way as the constant c is altered.