Question

Find the rate of change at(see picture)Write the exact answer. Do not round.

Answer 1

the Given:

The expression is given as,

`y=\sqrt[]{x+7}\text{ . . . . . (1)}`

The value of dx/dt is,

`(dx)/(dt)=7\text{ . . . . . . (2)}`

The objective is to find dy/dt at y = 6.

Step-by-step explanation:

Substitute y = 6 in equation (1),

`\begin{gathered} y=\sqrt[]{x+7} \\ 6=\sqrt[]{x+7}\text{ . . . . (3)} \end{gathered}`

To find dy/dt:

Differentiate equation (1) for t.

`\begin{gathered} \frac{dy}{d\text{t}}=\frac{dy}{d\text{x}}*\frac{d\text{x}}{dt} \\ =\frac{d}{d\text{x}}\sqrt[]{x+7}*\frac{d\text{x}}{\mathrm{d}t} \\ =\frac{1}{2\sqrt[]{x+7}}*\frac{d\text{x}}{\mathrm{d}t}\text{ . . . . .(4)} \end{gathered}`

Substitute the value of equations (2) and (3) in equation (4).

`\begin{gathered} \frac{d\text{ y}}{d\text{ t}}=\frac{1}{2\sqrt[]{x+7}}*\frac{d\text{ x}}{d\text{ t}} \\ =(1)/(2*6)*7 \\ =(7)/(12) \end{gathered}`

Hence, the value of the rate of change is 7/12.