Question

Find the rate of change. Write the exact answer do not round

Answer 1

To solve this exercise, we need to use the chain rule, shown below

$\frac{dy}{dt}=\frac{d}{dx}y\cdot\frac{d}{dt}x$

Then, in our case

$\frac{d}{dx}y=\frac{d(\sqrt[]{x+1})}{dx}=\frac{d((x+1)^{\frac{1}{2}})}{dx}=\frac{1}{2}(x+1)^{-\frac{1}{2}}$

Therefore,

$\frac{d}{dt}y=\frac{1}{2\sqrt[]{x+1}}\cdot3$

Finally, evaluate dy/dt in y=5, as shown next.

$\begin{gathered} y=5 \\ \text{and} \\ y=\sqrt[]{x+1} \\ \Rightarrow\sqrt[]{x+1}=5 \\ \Rightarrow\frac{d}{dt}y(5)=\frac{1}{2\cdot5}\cdot3=\frac{3}{10} \end{gathered}$

The answer is dy/dt=3/10, when y=5.