Use the definition of a derivative to find f^'(x) and f^''(x) . Then graph f, f^', and f^'' on a

step 1 F of x is equal to x cubed minus 3x. And we want to find F prime of x first of all. So that's go

Use the definition of a derivative to find f^'(x) and...

Use the definition of a derivative to find $f^{\prime}(x)$ and
$f^{\prime \prime}(x) .$ Then graph $f, f^{\prime},$ and $f^{\prime \prime}$ on a common screen and check to see if your answers are reasonable.
$$f(x)=x^{3}-3 x$$

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Key Concepts

Limit Definition of the Derivative

The derivative is defined as the limit of the difference quotient, which involves evaluating the limit as the increment approaches zero. This concept is foundational in calculus as it provides the formal way to compute the instantaneous rate of change of a function, which is critical for understanding the behavior of the function at any given point.

Second Derivative

The second derivative is the derivative of the first derivative and is computed by applying the definition of the derivative again. It provides information on the concavity of the original function and is essential for understanding acceleration, inflection points, and the overall shape of the graph of the function.

Graphical Analysis of Functions and Their Derivatives

Graphing a function along with its first and second derivatives allows for a visual examination of the function’s behavior. This includes verifying critical points, understanding intervals of increase and decrease, and identifying concavity and inflection points. Such visual checks are important tools for confirming the analytical results of differentiation.

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