Problem 46 - College Algebra and Calculus: An Applied Approach

Question

find \(f^{\prime}(x)\). $$ f(x)=\frac{-6 x^{3}+3 x^{2}-2 x+1}{x} $$

Step-by-Step Solution

Answer

The derivative of the given function \(f(x)\) is \(f^{\prime}(x)=-12 x+3-\frac{1}{x^{2}}\)

Step 1: Separate the function into individual terms

Separate the given function: \(f(x)=\frac{-6 x^{3}+3 x^{2}-2 x+1}{x}\) into individual terms to facilitate easy calculations. This gives us: \(f(x)=-6 x^{2}+3 x-2+ \frac{1}{x}\)

Step 2: Apply Power Rule

Apply the power rule which states that the derivative of \(x^{n}\) is \(n*x^{n-1}\) to each term: \[f^{\prime}(x)=-12 x+3-0-\frac{1}{x^{2}}\]

Step 3: Simplify the function

Simplify the above expression to get the final derivative of the function: So, \(f^{\prime}(x)=-12 x+3-\frac{1}{x^{2}}\)

Key Concepts

Power Rule

One of the most fundamental techniques for finding the derivative of a polynomial term is the Power Rule. This rule provides a straightforward method to calculate derivatives and is expressed as follows: if you have a function like \(x^n\), its derivative is \(n \cdot x^{n-1}\). This method is particularly useful because it allows us to quickly find the rate at which a function changes at any given point.

In our exercise, the Power Rule is applied to several terms:

• For \(-6x^3\), the derivative is calculated by multiplying the exponent 3 by the coefficient -6, resulting in \(-18x^{3-1}\) or \(-18x^2\).
• For \(3x^2\), it becomes \(6x^{1}\) or simply \(6x\).
• For \(-2x\), the derivative is \(-2\) since the exponent 1 becomes 0, resulting in \(-2\). The derivative of a constant term like 1 is zero, as constants do not change.
• The term \(\frac{1}{x}\) is rewritten as \(x^{-1}\), and applying the Power Rule gives us \(-1x^{-2}\) or \(-\frac{1}{x^2}\).

Understanding the Power Rule is essential for simplifying expressions and finding derivatives quickly.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number exponents with coefficients. They are a core part of calculus due to their straightforward composition and ease of manipulation.

In this exercise, the original function \(f(x)=\frac{-6x^{3}+3x^{2}-2x+1}{x}\) includes a polynomial in the numerator. By separating the terms, the function can be more easily differentiated term-by-term. This step involves dividing each term in the polynomial by \(x\), converting it into separate terms as follows:

• \(-6x^3/x\) becomes \(-6x^2\).
• \(3x^2/x\) simplifies to \(3x\).
• \(-2x/x\) results in \(-2\).
• The constant \(1/x\) remains as \(\frac{1}{x}\), which is vital as it behaves differently when deriving due to its negative exponent when rewritten as \(x^{-1}\).

Recognizing these components helps in applying rules like the Power Rule and understanding the behavior and structure of polynomial functions.

Rational Functions

Rational functions are defined as the ratio of two polynomial expressions. They are more complex than simple polynomials because they can include variables in the denominator, potentially causing undefined expressions when the denominator equals zero.

In this exercise, the original function \(f(x)=\frac{-6x^{3}+3x^{2}-2x+1}{x}\) is an example of a rational function since it includes a single polynomial divided by \(x\). By rewriting it as separate terms \(-6x^2 + 3x - 2 + \frac{1}{x}\), we effectively eliminate the complexity of having a variable in the denominator.

When differentiating rational functions, it's crucial to handle terms separately, particularly those with negative exponents or variables in the denominator. This entails recognizing terms like \(\frac{1}{x}\), which, once converted to \(x^{-1}\), highlight the importance of rules like the Power Rule, especially as they allow us to find derivatives for terms not easily handled by simple polynomial derivatives.

Understanding rational functions helps in calculating derivatives accurately and prepares you for more advanced calculus topics.