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In calculus, a derivative represents how a function changes as its input changes. It's like measuring the slope of a curve at any given point. When we talk about derivatives, especially in the context of critical numbers, we focus on points where this slope becomes zero or undefined. Finding the derivative of a function is a key step when you want to understand its behavior.

To find the derivative, you perform calculus operations that differentiate the function. In practical terms, think of it as finding a formula that tells you the rate at which your original function is changing. This knowledge is crucial when identifying critical points, as these points can signify local maxima, minima, or points of inflection, which are essential features of a function's graph.

The power rule is one of the most straightforward methods for finding derivatives. It's typically one of the first and most commonly used rules in calculus for speeding up the differentiation process. Here’s how it works:
- If you have a function in the form of \( x^n \), its derivative is \( nx^{n-1} \).- For example, the derivative of \( x^3 \) is \( 3x^2 \), since you bring down the exponent as a multiplier and then decrease the original exponent by one.

In the case of the function \( f(x) = x^3 - 27x \), apply the power rule to each term separately:
- For the \( x^3 \) term, the derivative is \( 3x^2 \).
- For the \( -27x \) term, it's simply \(-27\), because the derivative of any constant multiplied by \( x \) is that constant.

This gives you the derivative \( f'(x) = 3x^2 - 27 \). You’ve now translated the function into a slope description that helps identify critical points by setting it equal to zero.

Solving equations is like finding the unknown values that make the equation true. After deriving the function, the next step in identifying critical numbers is to set the derivative equal to zero. This will help find where the slope of the function’s graph is horizontal, indicating potential critical points.

For our derivative, \( 3x^2 - 27 = 0 \), the process involves a few clear steps:

• Firstly, simplify if possible, such as dividing every term by the same non-zero number. Here, divide by 3, resulting in \( x^2 - 9 = 0 \).
• Next, solve for \( x \). Recognize that \( x^2 - 9 \) is a difference of squares, \( (x-3)(x+3) = 0 \), offering two solutions: \( x = 3 \) and \( x = -3 \).

The solutions, \( x = 3 \) and \( x = -3 \), are the critical numbers of interest, as they show where the derivative of the function equals zero.

Factoring polynomials refers to expressing the polynomial as a product of its factors. This is a key skill when solving quadratic equations or higher-order polynomials. In our example, we needed to factor \( x^2 - 9 \).

Factoring can seem challenging, but a few strategies simplify it. Here, we use the rule for factoring a difference of squares. The general form is \( a^2 - b^2 = (a-b)(a+b) \). Apply this to \( x^2 - 9 \) as follows:

• Recognize \( x^2 - 9 \) fits this pattern, where \( a = x \) and \( b = 3 \)
• Factor it into \( (x-3)(x+3) \), each expression equaling zero separately provides the values of \( x \).
• This gives the critical numbers \( x = 3 \) and \( x = -3 \).

This method greatly simplifies finding critical numbers and understanding the behavior of polynomial functions.