Evaluating a Second Derivative In Exercises 89-92 , evaluate...

Evaluating a Second Derivative In Exercises $89-92$ , evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result.
$$h(x)=\frac{1}{9}(3 x+1)^{3},\left(1, \frac{64}{9}\right)$$

Key Concepts

Second Derivative

The second derivative is the derivative of the first derivative, which provides information about the concavity and acceleration of the function. It indicates how the rate of change of the function itself is changing, and it is done by differentiating the first derivative.

Chain Rule

The chain rule is a fundamental method in differentiation used for finding the derivative of composite functions. When a function is composed of an outer function and an inner function, the chain rule directs that you differentiate the outer function evaluated at the inner function, and then multiply by the derivative of the inner function.

Differentiation of Polynomial Functions

Differentiation of polynomial functions involves applying standard differentiation rules, such as the power rule, to each term of the polynomial. When dealing with composite polynomial functions, the chain rule is often applied to accurately compute the derivative.

Function Evaluation at a Specific Point

Evaluating a function or its derivatives at a specific point means substituting the given value into the function or its derivative. This process is used to determine the behavior or characteristics of the function, such as its slope or concavity, at that exact point.

Use of Computer Algebra Systems (CAS)

Computer Algebra Systems are tools designed to perform symbolic mathematics, providing exact analytical solutions to complex mathematical problems. They are particularly useful for verifying manual computations like derivatives, integrals, and solving differential equations, ensuring accuracy in the problem-solving process.

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