Problem 62 - Calculus Single Variable

Question

Suppose that \(p\) is a degree \(n\) polynomial, and \(q\) is a degree \(m\) polynomial. What is the degree of \((p \cdot q)^{\prime} ?\) Of \((p \circ q)^{\prime} ?\) Of the numerator and denominator of \((p / q)^{\prime}\) (before performing any cancellations)?

Step-by-Step Solution

Answer

The degree of \((p \cdot q)^{\prime}\) is \(n+m-1\), for \((p \circ q)^{\prime}\), it's \(n \times m\), and for \((p/q)^{\prime}\), numerator is \(n+m-1\) and denominator is \(2m\).

Step 1: Degree of (p \cdot q)^{\prime}

The product of two polynomials, \(p\) and \(q\), with degrees \(n\) and \(m\) respectively, has a degree of \(n + m\). The derivative of a polynomial reduces the degree by 1, so the degree of \((p \cdot q)^{\prime}\) is \(n + m - 1\).

Step 2: Degree of (p \circ q)^{\prime}

For the composition \(p \circ q\), the degree is given by the product of the degrees of \(p\) and \(q\), so it is \(n \times m\). The chain rule says the derivative of a composition is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This rule usually retains the degree, so the degree of \((p \circ q)^{\prime}\) will be \(n \times m\).

Step 3: Degree of Numerator of (p / q)^{\prime}

Using the quotient rule, \((p/q)^{\prime} = \frac{q \cdot p^{\prime} - p \cdot q^{\prime}}{q^2}\). The numerator is \(q \cdot p^{\prime} - p \cdot q^{\prime}\). The degree of \(q \cdot p^{\prime}\) is \(m + n - 1\) and the degree of \(p \cdot q^{\prime}\) is \(n + m - 1\). So, the degree of the numerator is the same as these, \(n + m - 1\).

Step 4: Degree of Denominator of (p / q)^{\prime}

The denominator, \(q^2\), has a degree of \(2m\). This is twice the degree of the polynomial \(q\). As there are no cancellations considered, the degree does not reduce.

Key Concepts

Polynomial Degree

The degree of a polynomial refers to the highest power of the variable present in the polynomial expression. For example, in the polynomial \(3x^4 + 2x^2 + x\), the degree is 4 because the term with the highest exponent is \(x^4\). The degree of a polynomial provides important information about its behavior, particularly how it behaves as the variable approaches infinity. In the context of calculus, understanding the degree is crucial when exploring operations such as derivation. Derivatives often reduce the power of each term, generally reducing the degree of a polynomial by one. When dealing with more complex expressions that involve multiple polynomials, such as products or compositions, the degrees interact in particular ways that are key to solving calculus problems efficiently.

Product Rule

The product rule is an essential concept in calculus used when finding the derivative of the product of two functions. If you have two functions \( f(x) \) and \( g(x) \), the product rule states that the derivative of their product is given by:

• \((f \, g)' = f' \, g + f \, g'\).

This formula may seem a bit complex at first, but it essentially combines the effects of each function changing while the other remains constant and adds them together. For polynomials, the degree of the product initially is the sum of the individual degrees. When you apply the product rule, the calculation of the resulting polynomial's derivative reduces it by one due to the differentiation. Thus, you'll often find problems asking for the derivative of a product of polynomials require you to consider the degree after applying the rule.

Quotient Rule

The quotient rule helps in finding the derivative of a division between two functions, denoted as \(\frac{f(x)}{g(x)}\). The quotient rule is expressed by:

• \((\frac{f}{g})' = \frac{g \, f' - f \, g'}{g^2}\).

This rule is critical when dealing with rational expressions, where polynomials are divided by each other. Computing this derivative involves paying close attention to both the numerator and denominator. The degree of the original quotient is altered due to the nature of differentiation where the numerator degree remains \(n + m - 1\), representing each polynomial’s interaction, while the denominator degree becomes \(2m\) since \(g(x)\) is squared. Understanding these degree changes ensures an accurate assessment of a polynomial's behavior after differentiation.

Chain Rule

The chain rule is a fundamental calculus principle used when differentiating composite functions. Composites involve one function embedded within another, such as \( p(q(x)) \). The chain rule provides a way to differentiate these functions by expressing the resulting derivative as:

• \((f(g(x)))' = f'(g(x)) \, g'(x)\).

In polynomial calculus, this rule plays a critical role in simplifying the process. For example, if \(p(x)\) is a degree \(n\) polynomial and \(q(x)\) is a degree \(m\) polynomial, the composition \(p \, \circ \, q\) results in a degree of \(n \, \times \, m\). When finding the derivative using the chain rule, the degree of the resulting derivative remains unchanged. This is one reason why the chain rule can be particularly efficient: it maintains the degree through the combination of functions, keeping polynomial manipulation simpler.