Chapter 4 - Calculating the Derivative - 4.2 Derivatives of Products and Quotients - 4.2 Exercises - Page 216: 22

$$ r^{\prime}(t) =\frac{3-2t}{2 \sqrt{t}(2 t+3)^{2}} $$

Since $$ r(t)=\frac{\sqrt{t}}{2 t+3} $$ Then \begin{align*} r^{\prime}(t) &=\frac{(2 t+3) \frac{1}{2 \sqrt{t}}-2 \sqrt{t}}{(2 t+3)^{2}} \\ &=\frac{\frac{1}{2 \sqrt{t}}(2 t+3-2 t)}{(2 t+3)^{2}} \\ &=\frac{3-2t}{2 \sqrt{t}(2 t+3)^{2}} \end{align*}