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When finding derivatives of functions that are products of two or more other functions, we utilize a key tool known as the product rule. In calculus, the product rule is essential for differentiating expressions where multiple variable expressions are multiplied together. It's stated as follows:
Suppose you have two functions, say \( u(x) \) and \( v(x) \), the derivative of their product \( u(x)v(x) \) with respect to \( x \) is given by:
\[ u'(x)v(x) + u(x)v'(x) \].
For the function in the exercise \( f(x)=\big(2x^3+5x\big)\big(x-3\big)\big(x+2\big) \), we initially break it down into three constituent functions \( u(x) = (2x^3 + 5x) \), \( v(x) = (x-3) \), and \( w(x) = (x+2) \). Following the product rule for three functions, the derivative \( f'(x) \) is the sum of each function multiplied by the derivative of the other two. This concept ensures that every component of the original function is accounted for in the differentiation process, allowing students to tackle even the most complex product of functions.
When applying the product rule, it's important to proceed systematically, differentiating one function at a time and then summing up the results. This process prevents any terms from being overlooked and guarantees a thorough and accurate derivative.

The power rule is one of the most basic and frequently used rules for taking derivatives in calculus. It applies when you are differentiating polynomials or any functions that can be expressed as a variable raised to a power. The power rule states that if you have a function of the form \( x^n \), where \( n \) is any real number, then its derivative is \( nx^{n-1} \).
Using this rule, the derivatives of the individual functions from our original exercise, namely \( u(x)=(2x^{3}+5x) \), \( v(x)=(x-3) \), and \( w(x)=(x+2) \), become \( u'(x)=6x^{2}+5 \), as 3 is reduced by 1 for \( 2x^{3} \) part and the derivative of \( 5x \) becomes 5, while the derivatives for \( v'(x) \) and \( w'(x) \) both simplify to 1, since they are effectively \( x^1 \) and the power rule for \( x^1 \) is simply the coefficient, which is 1 in both cases.
The power rule greatly simplifies the process of differentiation by reducing complex polynomial expressions to more manageable terms quickly. It's a foundational technique that allows students to differentiate a wide variety of algebraic functions swiftly and efficiently, eliminating the need for lengthy calculations and directly leading to the core of the problem.

Algebraic functions form a broad category of functions that include polynomials, rational functions, and root functions, among others. They are defined by algebraic expressions using operations of addition, subtraction, multiplication, division, and taking roots within the domain of real numbers. These functions often represent the relationship between variables in mathematical models and are frequently encountered in calculus.
In the context of the exercise \( f(x)=\big(2x^3+5x\big)\big(x-3\big)\big(x+2\big) \), we're working with an algebraic function that is a product of three simpler algebraic functions. Understanding the nature of these component functions helps when applying differentiation rules, like the product and power rules.
Algebraic functions are prime candidates for derivative computations using established calculus rules. By breaking down complex algebraic functions into their elementary components or recognizing their standard forms, students can apply the product and power rules with confidence. The ability to manipulate these functions algebraically before taking the derivative also simplifies the entire differentiation process, leading to more straightforward solutions.

The process of derivative simplification involves reducing the derived expression to its simplest form. This can include expanding products, combining like terms, and canceling common factors. Simplification is often the final step in the differentiation process and is crucial for making the resulting derivative more interpretable and easier to use in further applications, such as solving equations or graphing functions.
When we reach the simplification stage in our exercise, after having applied the product and power rules, we're left with a derivative that might seem complex at first glance: \( f'(x) = (6x^2+5)(x-3)(x+2) + (2x^3+5x)(1)(x+2) + (2x^3+5x)(x-3)(1) \). The task now is to expand the products and combine like terms. For example, we will multiply out the factors of each term, add or subtract similar terms, and align the resulting expression in decreasing powers of \( x \).
This step is where a deep understanding of algebra pays off, allowing for the simplification of the derivative to its most reduced form. Simplifying the derivative not only provides clarity but also prepares the expression for any subsequent calculations that may be necessary, such as finding critical points or solving optimization problems.