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The derivative is a fundamental concept in calculus, representing the rate at which a function changes at any given point. Calculating the derivative of a function involves finding how the output value of the function changes with a slight change in the input value. It provides crucial information about the behavior of functions, such as whether they are increasing or decreasing.

To find the derivative, we often use rules like the power rule, the product rule, or the chain rule, depending on the complexity of the function. In the case of the function \( f(x) = x^3 - 5x \), its derivative is found using the power rule, leading to \( \frac{df}{dx} = 3x^2 - 5 \). This derivative \( \frac{df}{dx} \) tells us the rate of change at any point \( x \) along the curve.

The rate of change is a measure of how quickly a function's value changes as the input changes. It is closely tied to the concept of the derivative in calculus.

When we calculate the derivative of a function, we actually determine the instantaneous rate of change at a specific point. For example, evaluating the derivative \( \frac{df}{dx} = 3x^2 - 5 \) at \( x = 2 \) gives us a rate of change of \( 7 \). This positive value indicates that at \( x = 2 \), the function \( f(x) = x^3 - 5x \) is increasing. Understanding the rate of change allows us to predict the behavior of functions and graphs, determine trends, and solve practical problems.

A function is increasing on an interval if, as \( x \) increases, the function value \( f(x) \) also increases. Conversely, a function is decreasing if \( f(x) \) decreases as \( x \) increases within an interval.

To determine whether a function is increasing or decreasing, we examine its derivative. If \( \frac{df}{dx} > 0 \) for a certain interval, the function is increasing within that region. If \( \frac{df}{dx} < 0 \), the function is decreasing. For example, for \( f(x) = x^3 - 5x \), the derivative \( 3x^2 - 5 \) is positive at \( x = 2 \), meaning the function is increasing at this point.

• When \( x \) ranges between approximately \( -1.29 \) and \( 1.29 \), the derivative \( 3x^2 - 5 \) is negative, indicating that the function decreases in this interval.

By analyzing these changes, we discover where the function rises or falls and make informed predictions or interpretations accordingly.

Critical points occur where the derivative of a function is zero or undefined. These points are vital in determining the function's behavior, as they often signal a change from increasing to decreasing (or vice versa) or indicate points where the function reaches local maxima or minima.

To find critical points of \( f(x) = x^3 - 5x \), set the derivative \( \frac{df}{dx} = 3x^2 - 5 \) to zero and solve for \( x \). Solving \( 3x^2 - 5 = 0 \) gives us the critical points at \( x = \pm\sqrt{\frac{5}{3}} \). By examining these critical points, we can further analyze the behavior of the function. In these regions, the function may transition from increasing to decreasing or vice versa, providing crucial insights into the curve's overall shape.
Understanding critical points helps in sketching the graph of a function and predicting points of interest, such as peaks and troughs in the graph.