the Given:
The expression is given as,
[tex]y=\sqrt[]{x+7}\text{ . . . . . (1)}[/tex]
The value of dx/dt is,
[tex]\frac{dx}{dt}=7\text{ . . . . . . (2)}[/tex]
The objective is to find dy/dt at y = 6.
Explanation:
Substitute y = 6 in equation (1),
[tex]\begin{gathered} y=\sqrt[]{x+7} \\ 6=\sqrt[]{x+7}\text{ . . . . (3)} \end{gathered}[/tex]
To find dy/dt:
Differentiate equation (1) for t.
[tex]\begin{gathered} \frac{dy}{d\text{t}}=\frac{dy}{d\text{x}}\times\frac{d\text{x}}{dt} \\ =\frac{d}{d\text{x}}\sqrt[]{x+7}\times\frac{d\text{x}}{\mathrm{d}t} \\ =\frac{1}{2\sqrt[]{x+7}}\times\frac{d\text{x}}{\mathrm{d}t}\text{ . . . . .(4)} \end{gathered}[/tex]
Substitute the value of equations (2) and (3) in equation (4).
[tex]\begin{gathered} \frac{d\text{ y}}{d\text{ t}}=\frac{1}{2\sqrt[]{x+7}}\times\frac{d\text{ x}}{d\text{ t}} \\ =\frac{1}{2\times6}\times7 \\ =\frac{7}{12} \end{gathered}[/tex]
Hence, the value of the rate of change is 7/12.
