Question

$$\begin{gathered} \mathrm{Linda}\mathrm{can}\mathrm{sell}12\mathrm{magazine}\mathrm{subscriptions}\mathrm{per}\mathrm{week}\mathrm{and}\mathrm{makes}\backslash (4.00 \\ \mathrm{for}\mathrm{each}\mathrm{magazine}\mathrm{subscription}\mathrm{she}\mathrm{sells}.\mathrm{Obviously}\mathrm{this}\mathrm{means}\mathrm{that} \\ \mathrm{Linda}\mathrm{will}\mathrm{make}\left(12\right)(\backslash )4.00)=\$48.00\mathrm{per}\mathrm{week}\mathrm{from}\mathrm{magazine}\mathrm{subscriptions}. \\ \mathrm{Explain}\mathrm{this}\mathrm{result}\mathrm{mathematically},\mathrm{using}\mathrm{mathematical}\mathrm{notation}\mathrm{and}\mathrm{the}\mathrm{chain}\mathrm{rule}. \end{gathered}$$

Short answer

$$\begin{gathered} \mathrm{In}\mathrm{mathematical}\mathrm{notations} \\ \mathrm{The}\mathrm{amount}\mathrm{she}\mathrm{make}\mathrm{per}\mathrm{week}\mathrm{is}, \\ \frac{d\mathrm{D}}{d\mathrm{t}}=\frac{d\mathrm{D}}{d\mathrm{m}}\times \frac{d\mathrm{m}}{d\mathrm{t}} \\ =4\times 12 \\ =48. \end{gathered}$$

Step-by-step solution

Step 1. Given Information

$$\begin{gathered} \mathrm{The}\mathrm{number}\mathrm{of}\mathrm{magazine}\mathrm{subscriptions}\mathrm{per}\mathrm{week}\mathrm{Linda}\mathrm{can}\mathrm{sell}\mathrm{is}12\mathrm{the}\mathrm{earning}\mathrm{is}\$4. \\ \mathrm{per}\mathrm{magazine}\mathrm{is}\frac{d\mathrm{D}}{d\mathrm{m}}=4. \end{gathered}$$

Step 2. Solution

$$\begin{gathered} \mathrm{The}\mathrm{amount}\mathrm{she}\mathrm{make}\mathrm{per}\mathrm{week}=\left(12\right)(\$4)=\$48. \\ \mathrm{In}\mathrm{mathematical}\mathrm{notations} \\ \mathrm{The}\mathrm{amount}\mathrm{she}\mathrm{make}\mathrm{per}\mathrm{week}\mathrm{is}, \\ \frac{d\mathrm{D}}{d\mathrm{t}}=\frac{d\mathrm{D}}{d\mathrm{m}}\times \frac{d\mathrm{m}}{d\mathrm{t}} \\ =4\times 12 \\ =48. \end{gathered}$$