SOLUTION:
Step 1:
In this question, we are given the following:
Use the limit definition of the derivative to find the instantaneous rate of change of
Step 2:
The details of the solution are as follows:
An alternative way to find the use the limit definition of the derivative to find the instantaneous rate of change of:
[tex]f(x)\text{ =}\sqrt[]{4x+8\text{ }}\text{ at x = 4}[/tex]
Let us take the derivative of the function and we have that:
[tex]\begin{gathered} f(x)\text{ = }\sqrt[]{4x+8}=(4x+8)^{\frac{1}{2}} \\ \end{gathered}[/tex][tex]\begin{gathered} f^I(x)\text{ = }\frac{1}{2}(4x+8)^{-\frac{1}{2}}(4) \\ f^I(x)=2(4x+8)^{-\frac{1}{2}},\text{ when x = 4, we have that:} \\ f^I(4)=2(4(4)+8)^{-\frac{1}{2}} \end{gathered}[/tex][tex]\begin{gathered} f^I(4)=2(16+8)^{-\frac{1}{2}} \\ f^I(4)\text{ = 2 x }\frac{1}{\sqrt[]{24}} \end{gathered}[/tex][tex]f^I(4)\text{ =}\frac{2}{\sqrt[]{24}}[/tex][tex]f^I(4)=\frac{2}{\sqrt[]{24}}=\frac{2}{\sqrt[]{4}X\sqrt[]{6}}=\frac{2}{2X\sqrt[]{6}}=\frac{1}{\sqrt[]{6}}[/tex][tex]f^I(4)\text{ =}\frac{1}{\sqrt[]{6}}[/tex]
