Question

The reaction of the body to a dose of medicine can sometimes be represented by an equation of the form R

​, 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
. This​ derivative, as a function of​ M, is called the sensitivity of the body to the medicine.

Answer 1

`\frac{\text{d}R}{\text{d}M}=-M^2+CM`

Explanation:

The reaction of the body to a dose of medicine can sometimes be represented by an equation of the form

`R=M^2\left((C)/(2)-(M)/(3)\right)`

where C is a positive constant and M is the amount of medicine absorbed in the blood.

To find dR/dM, differentiate R with respect to M using the product rule of differentiation.

`\boxed{\begin{minipage}{5.5 cm}\underline{Product Rule for Differentiation}\\\\If $y=uv$ then:\\\\$\frac{\text{d}y}{\text{d}x}=u\frac{\text{d}v}{\text{d}x}+v\frac{\text{d}u}{\text{d}x}$\\\end{minipage}}`

`\textsf{Let}\;\;u=M^2 \implies \frac{\text{d}u}{\text{d}M}=2M`

`\textsf{Let}\;\;v=(C)/(2)-(M)/(3) \implies \frac{\text{d}v}{\text{d}M}=-(1)/(3)`

(Remember that C is a positive constant, and the derivative of a constant is zero).

Substituting the values into the product rule formula, we get:

`\begin{aligned}\frac{\text{d}R}{\text{d}M}&=u\frac{\text{d}v}{\text{d}M}+v\frac{\text{d}u}{\text{d}M}\\\\&=M^2 \left(-(1)/(3)\right)+\left((C)/(2)-(M)/(3)\right)(2M)\\\\&=-(1)/(3)M^2+2M\left((C)/(2)-(M)/(3)\right)\\\\&=-(1)/(3)M^2+CM-(2)/(3)M^2\\\\&=-M^2+CM\end{aligned}`