`y = sqrt(x^2 + 8x), (1,3)` Find and evaluate the derivative of the function at the given point. Use a graphing utility to verify your result.

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Chapter 2, 2.4 - Problem 65 - Calculus of a Single Variable (10th Edition, Ron Larson).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the derivative of the function, using the chain rule, such that:

`y' = ((x^2 + 8x)^(1/2))' => y' = ((1/2)(x^2 + 8x)^(1/2 - 1))(x^2 + 8x)'`

`y' = ((1/2)(x^2 + 8x)^(-1/2))(2x + 8)`

`y' = (2x + 8)/(2sqrt(x^2 + 8x))`

Factoring out 2, yields:

`y' = (2(x + 4))/(2sqrt(x^2 + 8x))`

Reducing by 2 yields:

`y' = (x + 4)/(sqrt(x^2 + 8x))`

Now, you need to evaluate the value of derivative at x = 1:

`y' = (1 + 4)/(sqrt(1^2 + 8))`

`y' = (5)/(sqrt(9))`

`y' = 5/3`

Hence, evaluating the derivative of the function yields `y' = (x + 4)/(sqrt(x^2 + 8x))` and evaluating the value of derivative at x = 1, yields `y' = 5/3` .

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