`h(x) = (1/9)(3x + 1)^3, (1,(64/9))` Evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result.

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Chapter 2, 2.4 - Problem 91 - 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 first to evaluate the first derivative of the function, using chain rule, such that:

`h'(x) = ((1/9)(3x + 1)^3)'`

`h'(x) = ((3/9)(3x + 1)^2)*(3x + 1)'`

`h'(x) = ((1/3)(3x + 1)^2)*(3)`

`h'(x) = ((3/3)(3x + 1)^2)`

`h'(x) = (3x + 1)^2`

Now you may evaluate the second derivative, using the quotient rule:

`h''(x) = ((3x + 1)^2)'`

`h''(x) = (2(3x + 1))*(3x + 1)'`

`h''(x) = (2(3x + 1))*3`

`h''(x) = (6(3x + 1))`

You need to evaluate the value of the second derivative at x = 1, such that:

`h''(1) = (6(3*1 + 1))`

`h''(1) = 24`

Hence, evaluating the second derivative expression yields `h''(x) = (6(3x + 1))` and evaluating the value of the second derivative at x = 1, yields `h''(1) = 24.`

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