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

`(dy)/(dt)=(d)/(dx)y\cdot(d)/(dt)x`

Then, in our case

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

Therefore,

`(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(d)/(dt)y(5)=(1)/(2\cdot5)\cdot3=(3)/(10) \end{gathered}`

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