## Calculus with Applications (10th Edition)

A demand function is given by \begin{aligned} q &=D(p)\\ &=30\left(5-\frac{p}{\sqrt{p^{2}+1}}\right) \\ &=150-\frac{30 p}{\sqrt{p^{2}+1}} \\ &=150-\frac{30 p}{\left(p^{2}+1\right)^{1 / 2}}. \end{aligned} The rate of change in the demand for the product per unit change in price \begin{aligned} \frac{d q}{d p} &=0-\left(\frac{\left(p^{2}+1\right)^{1 / 2} D_{p}\left(30_{p}\right)-\left(30_{q}\right)\left(D_{p}\left(p^{2}+1\right)^{1 / 2}\right)}{\left[\left(p^{2}+1\right)^{1 / 2}\right]^{2}}\right.\\ &=-\left[\frac{\left(p^{2}+1\right)^{1 / 2}(30)-(30 p)\left(\frac{1}{2}\right)\left(p^{2}+1\right)^{-1 / 2}(2 p)}{\left[p^{2}+1\right)}\right] \\ &=-\left[\frac{\left.\left(p^{2}+1\right)^{1 / 2}(30)-(30 p)(p)\left(p^{2}+1\right)^{-1 / 2}\right)}{\left(p^{2}+1\right)}\right] \\ &=-\frac{-30\left(p^{2}+1\right)^{-1 / 2}\left(\left[p^{2}+1\right]-p^{2}\right)}{\left(p^{2}+1\right)} | \\ &=-\frac{-30\left(p^{2}+1\right)^{-12}(1)}{\left(p^{2}+1\right)} \\ &=-\frac{30}{\left(p^{2}+1\right)^{3 / 2}} \end{aligned}
A demand function is given by \begin{aligned} q &=D(p)\\ &=30\left(5-\frac{p}{\sqrt{p^{2}+1}}\right) \\ &=150-\frac{30 p}{\sqrt{p^{2}+1}} \\ &=150-\frac{30 p}{\left(p^{2}+1\right)^{1 / 2}}. \end{aligned} The rate of change in the demand for the product per unit change in price \begin{aligned} \frac{d q}{d p} &=0-\left(\frac{\left(p^{2}+1\right)^{1 / 2} D_{p}\left(30_{p}\right)-\left(30_{q}\right)\left(D_{p}\left(p^{2}+1\right)^{1 / 2}\right)}{\left[\left(p^{2}+1\right)^{1 / 2}\right]^{2}}\right.\\ &=-\left[\frac{\left(p^{2}+1\right)^{1 / 2}(30)-(30 p)\left(\frac{1}{2}\right)\left(p^{2}+1\right)^{-1 / 2}(2 p)}{\left[p^{2}+1\right)}\right] \\ &=-\left[\frac{\left.\left(p^{2}+1\right)^{1 / 2}(30)-(30 p)(p)\left(p^{2}+1\right)^{-1 / 2}\right)}{\left(p^{2}+1\right)}\right] \\ &=-\frac{-30\left(p^{2}+1\right)^{-1 / 2}\left(\left[p^{2}+1\right]-p^{2}\right)}{\left(p^{2}+1\right)} | \\ &=-\frac{-30\left(p^{2}+1\right)^{-12}(1)}{\left(p^{2}+1\right)} \\ &=-\frac{30}{\left(p^{2}+1\right)^{3 / 2}} \end{aligned}