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Derivatives are a fundamental tool in calculus, widely used to determine how a function changes at any point. The derivative of a function gives us the rate at which the function's value is changing with respect to a change in its input. When you take the derivative, you are essentially finding the slope of the tangent line to the curve at any given point.

To find the derivative of a polynomial function, like the one in the original exercise, you apply the power rule to each term individually. This process involves a simple formula where you bring the exponent in front as a coefficient and then subtract one from the exponent.

• The function in question is a product of polynomials, which means initially you'll expand it, and then find the derivative of each term.
• The resulting expression provides a new polynomial that depicts the rates of change for each term at different points.

Polynomial functions are expressions made up of variables and coefficients, involving terms that are added, subtracted, or multiplied together. They have the form \( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \).

Polynomials are versatile and appear frequently in different areas of mathematics due to their simple structure and the ease with which they can be manipulated. In the given problem, the function is expressed as \( f(x) = x^3(x-5)^2 \), which initially appears as a product of expressions.

By expanding this expression, we convert it into a more straightforward polynomial form, \( x^5 - 10x^4 + 25x^3 \). This makes differentiation easier, allowing us to apply familiar rules to find critical numbers as needed.

The quadratic formula is a powerful tool used to solve quadratic equations—equations that can be rearranged into the standard form \( ax^2 + bx + c = 0 \). The formula for the solutions is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula provides an efficient way to find the roots of any quadratic equation, without needing to manually factor or graph the equation.

In our exercise, after setting the derivative to zero and factoring, we end up with the quadratic expression \( x^2 - 8x + 15 \). Applying the quadratic formula here, we found that the critical points came at \( x = 5 \) and \( x = 3 \), in addition to \( x = 0 \) derived earlier. This shows how the quadratic formula helps solve for unknown values, even within more complex polynomial scenarios.

The power rule is crucial when finding the derivative of polynomial terms. It is a straightforward method applied to functions of the form \( x^n \), where \( n \) is a constant exponent. The power rule states:\[ \frac{d}{dx} x^n = nx^{n-1} \]
This rule makes taking derivatives very manageable for polynomial functions because each term follows this pattern.

• For example, in the exercise provided, using the power rule on \( x^5 \), \( x^4 \), and \( x^3 \) results in derivatives of \( 5x^4 \), \( -40x^3 \), and \( 75x^2 \) respectively.
• By applying the power rule to each term in the expanded polynomial, you transform the original function into its derivative form, enabling the identification of critical numbers.