Problem 42 - College Algebra and Calculus: An Applied Approach

Question

find \(f^{\prime}(x)\). $$ f(x)=\left(3 x^{2}-5 x\right)\left(x^{2}+2\right) $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = (3x^{2} - 5x)(x^{2} + 2)\) is \(f^{\prime}(x) = 12x^{3} -15x^{2} + 12x - 10\).

Step 1: Find \(f'(x)\) for the first part of the product

Take the derivative of the first function representing \(f(x)\), which is \(3x^{2} - 5x\). Differentiating \(3x^{2}\) gives \(6x\), and differentiating \(-5x\) gives \(-5\). This yields \(f'(x)\)= \(6x - 5\).

Step 2: Find \(g'(x)\) for the second part of the product

Next, take the derivative of the second function representing \(g(x)\), which is \(x^{2} + 2\). Differentiating \(x^{2}\) gives \(2x\) and differentiating \(2\), a constant, gives \(0\). This yields \(g'(x)\) = \(2x\).

Step 3: Apply the product rule

According to the product rule, the derivative of the entire function is given by \((f(x)g(x))' = f'(x)g(x)+ f(x)g'(x)\). Substitute the found values for \(f'(x)\), \(g'(x)\), \(f(x)\) and \(g(x)\) into this formula: \((f(x)g(x))' = (6x - 5)(x^{2} + 2) + (3x^{2} - 5x)(2x)\). Simplifying this gives \(f^{\prime}(x) = 6x^3 + 12x - 5x^{2} - 10 + 6x^{3} -10x^{2} = 12x^{3} -15x^{2} +12x - 10\).

Key Concepts

Product Rule

The product rule is a fundamental principle in calculus that helps in finding the derivative of a product of two functions. If you have two functions multiplied together, say \( u(x) \) and \( v(x) \), the product rule states that the derivative of their product is given by:

• \((uv)' = u'v + uv'\)

This means you take the derivative of the first function and multiply it by the second function, then add to it the first function multiplied by the derivative of the second function. Understanding this rule is crucial because it allows us to break down complex expressions involving products into more manageable parts, making differentiation much simpler. In our original problem, we applied the product rule to the expressions \((3x^2 - 5x)\) and \((x^2 + 2)\) to find the derivative effectively. Each derivative was computed separately and then brought together using this rule.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative represents the rate of change of the function with respect to a variable, usually \( x \). Differentiation helps us understand how a function behaves, whether it is increasing, decreasing, or has any turning points.
When differentiating polynomial functions, each term is differentiated individually. You follow the power rule, which involves bringing down the exponent as a multiplier and reducing the exponent by one. For example, differentiating \( 3x^2 \) results in \( 6x \) and differentiating \(-5x\) leads to \(-5\). With constant terms like \(2\), their derivative is \(0\) because constants do not change.

• Poly-exponent reduction: Reduce the power of the variable by one
• Multiplier application: Multiply the coefficient by the original exponent

Differentiation is a key part of calculus as it unlocks the understanding of dynamic changes within functional contexts. In practical scenarios, once the basic rules are applied, these concepts make complex derivatives accessible and straightforward.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of variables, each multiplied by a coefficient, plus a constant term. These functions are widely used due to their simplicity and versatility for modeling real-world situations. A typical polynomial function looks like this:\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \]where \( n \) is a non-negative integer, and \( a_n, a_{n-1},... \) are constants called coefficients.
In using derivatives with polynomial functions, you will often encounter tasks of determining how the output of these functions changes when there is a slight change in the input. Calculating derivatives for polynomials utilizes simple rules derived from the power rule, making them straightforward. In our original question, the functions \( 3x^2 - 5x \) and \( x^2 + 2 \) are examples of polynomial functions.

• Easy to differentiate: Simple differentiation rules apply
• Predictable behavior: Polynomials don't exhibit sudden spikes or erratic changes within their domain

Because of their algebraic structure, polynomial functions provide a strong foundation in calculus and are often the first real-world examples students encounter.