Question

The Body's Reaction to Medicine The reaction of the body to a dose of medicine can often be represented by an equation of the form $$R=M^{2}\left(\frac{C}{2}-\frac{M}{3}\right)$$ where \(C\) is a positive constant and \(M\) is the amount of medicine absorbed in the blood. If the reaction is a change in blood pressure, \(R\) is measured in millimeters of mercury. If the reaction is a change in temperature, \(R\) is measured in degrees, and so on. Find \(d R / d M\) . This derivative, as a function of \(M,\) is called the sensitivity of the body to medicine. In Chapter \(4,\) we shall see how to find the amount of medicine to which the body is most sensitive. Source: Some Mathematical Models in Biology, Revised Edition, December \(1967, \mathrm{PB}-202, \mathrm{p} .221\) distributed by N.T.S.. U.S. Department of Commerce.

Short answer

The derivative of the body's reaction to medicine, in terms of the dosage, \(M\), is given by \(d R / d M = M\left(C - \frac{2M}{3}\right) - \frac{M^{2}}{3}\). This represents the sensitivity of the body to the medicine.

Step-by-step solution

Identify the function to be differentiated

The function representing the body's reaction to medicine is \(R=M^{2}\left(\frac{C}{2}-\frac{M}{3}\right)\). This function needs to be differentiated with respect to \(M\).

Apply the product rule of differentiation

The function is an example of a product of two functions (i.e., \(M^{2}\) and \(\left(\frac{C}{2}-\frac{M}{3}\right)\)). For a product of two functions \(f(x) \cdot g(x)\), the derivative is given by \(f'(x) \cdot g(x) + f(x) \cdot g'(x)\). Applying the product rule, the derivative becomes \(2M\left(\frac{C}{2}-\frac{M}{3}\right) + M^{2}\left(-\frac{1}{3}\right)\).

Simplify the result

Solving the expression, the derivative of \(R\) with respect to \(M\) simplifies to \(d R / d M = M\left(C - \frac{2M}{3}\right) - \frac{M^{2}}{3}\).

Key concepts

Product Rule of Differentiation

Understanding the product rule of differentiation is essential in calculus when dealing with the derivative of products of two or more functions. This rule states that if we have two differentiable functions, say, u(x) and v(x), the derivative of their product is given by:

\[ (u \times v)' = u' \times v + u \times v' \]
In the given exercise, we applied this rule to the function representing the body's reaction to medicine, which is the product of \(M^2\) and \(\frac{C}{2} - \frac{M}{3}\). Through application of the product rule, we were able to find the derivative, \(dR/dM\), which is fundamental to understanding the sensitivity of the body to the medicine. The elucidation of this process provides an insight into the connection between the amount of medicine and its effect on the body.

Sensitivity to Medicine

Sensitivity to medicine is a vital concept in pharmacology, denoting how the body's response varies with changes in medication dosage. This measure is crucial for determining the optimal amount of medicine for effective treatment. In calculus, we determine this sensitivity by calculating the derivative of the body's reaction function with respect to the medicine, denoted by \(dR/dM\).

The higher the absolute value of this derivative, the more sensitive the body's response to changes in the medicine amount. In the exercise, finding this derivative provides a mathematical model to predict how the body will react as the medicine dosage changes—the cornerstone of personalized medicine.

Derivative of a Function

The derivative of a function at a particular point represents the rate at which the function's value changes with respect to changes in its input variable. It's a fundamental concept in calculus, often symbolized by \(f'(x)\) or \(\frac{df}{dx}\), for a function \(f(x)\).

The derivative provides information about the function's rate of change and can be used to determine where the function is increasing or decreasing, where it has extrema, and the behavior of the function's graph. In the context of medicine, as in our exercise, the derivative helps in understanding how quickly the body reacts to the amount of medicine in the bloodstream, and thus it guides dosing decisions.

Calculus Applications in Biology

Calculus, particularly differentiation, finds abundant applications in biology, aiding in the quantification, and analysis of dynamic systems. It is used to model the growth of populations, the spread of diseases, and to understand the changes in heart rate, metabolism, and other physiological parameters.

In our exercise, calculus is applied to model the body's reaction to medicine—a topic within pharmacokinetics. By differentiating the function that represents how the body responds to a drug, we can analyze how sensitivity varies with dosage. Such models are invaluable for predicting the outcomes of medication regimes and for optimizing treatments to enhance efficacy and minimize side effects.