a. Find the derivative of f at x. That is, find f^'(x). b. Find the slope of the tangent line to

step 1 In this problem, we are given f of x as x cubed minus 2. Now first we have to find the derivativ

A. Find the derivative of f at x. That is, find f^'(x). b....

a. Find the derivative of $f$ at $x$. That is, find $f^{\prime}(x)$.
b. Find the slope of the tangent line to the graph of $f$ at each of the two values of $x$ given to the right of the function.
$$f(x)=x^{3}-2 ; x=-1, x=1$$

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Key Concepts

Tangent Line

The tangent line to a function at a point is a straight line that touches the graph of the function at that point and has the same instantaneous rate of change (slope) as the function. Its slope is determined by the value of the derivative evaluated at that point.

Evaluating the Derivative at a Point

Once the general derivative function is found, evaluating it at a specific point gives the slope of the tangent line to the graph of the function at that particular point. This process is essential for understanding how the function behaves locally around a given value.

Derivative

The derivative of a function is a fundamental concept in calculus that measures the instantaneous rate of change of the function with respect to its variable. It is defined as the limit of the difference quotient as the interval approaches zero and gives the slope of the tangent line to the function at any point.

Power Rule

The power rule is a basic differentiation rule that allows for the quick computation of the derivative of functions in the form of x raised to a power. It states that if f(x) = x^n, then the derivative f'(x) is n*x^(n-1), which is particularly useful for differentiating polynomial functions.

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