Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y=(x+1)(x+2)(x+3)(x+4),\) then \(d^{5} y / d x^{5}=0\)

Short answer

The statement is true. The fifth derivative of the given polynomial function is indeed 0, because the degree of the function (4) is lower than 5.

Step-by-step solution

Recap Derivative Rule for Polynomial

The derivative of a polynomial function yields another function with the degree of the original polynomial reduced by one. Thus, if \(y=P(x)\) where P is a polynomial of degree n, the \(m^{th}\) derivative of y, denoted \(d^m y / dx^m,\) would be a polynomial of degree \(n-m\) if \(mn\).

Determine the Degree of the Polynomial

Given \(y=(x+1)(x+2)(x+3)(x+4)\), it is clear to see that y is a polynomial of degree 4, since it is a product of four linear (degree 1) terms.

Apply the Derivative Rule & Answer the Statement

Since the degree of \(y\) is 4 and it is less than 5, the fifth derivative of \(y\), represented by \(d^5 y / dx^5,\) should be equal to zero, according to the derivative rule mentioned in Step 1. Hence, the given statement is true.

Key concepts

Derivative Rules

Understanding derivative rules is essential for calculus students as it lays the foundation for differentiating functions effectively. Derivatives represent the rate at which a function is changing at any given point and are fundamental in fields like physics, engineering, and economics for modeling change.

For polynomials, the power rule is one of the most basic derivative rules. It states that the derivative of a monomial (a single term polynomial) is found by multiplying the coefficient by the power and then subtracting one from the power. For example, the derivative of \(x^n\) with respect to \(x\) is \(nx^{n-1}\).

When dealing with polynomial functions, which are sums of monomials, we apply the power rule term by term, thanks to the linearity of derivatives. This means that the derivative of a sum is the sum of the derivatives. Moreover, for products of functions like \(y=(x+1)(x+2)(x+3)(x+4)\), the product rule should be applied. However, in this case, fully expanding the product into a polynomial of degree 4 and then applying the power rule is more efficient.

Another important aspect to remember is the rule for higher-order derivatives. Each time we take a derivative, we're essentially applying the power rule again, decreasing the degree of the polynomial by one each time until we reach a constant. When the degree of the polynomial is less than the number of derivatives we're taking, the derivative is zero.

Degree of a Polynomial

The degree of a polynomial is determined by the largest exponent of its variable. It defines the highest amount of turns and intersections with the x-axis that a polynomial graph can have. Knowing the degree offers insight into the behavior of the polynomial at the extremes of \(x\) and is particularly useful when differentiating.

In the provided exercise, \(y = (x+1)(x+2)(x+3)(x+4)\), we have a polynomial that is written as a product of linear factors. When expanded, it will be a polynomial of degree 4 because the degree of a product of functions is the sum of the degrees of the individual functions. Therefore, these factors, each of degree one, add up to a polynomial of degree 4.

This concept is crucial when dealing with derivatives because the degree informs us how many times we can derive a function before it reduces to zero. For instance, a polynomial of degree 4 can be differentiated up to four times with non-zero results. After that, subsequent derivatives will yield zero, as there is no \(x\)-term left to derive.

Higher-Order Derivatives

Higher-order derivatives refer to the repeated application of the differentiation process. While the first derivative represents the rate of change, each subsequent derivative provides information on the curvature and more complex patterns of change within a function.

In the context of the given problem, we seek the fifth derivative of a fourth-degree polynomial. As we know, taking the first derivative of a fourth-degree polynomial gives us a third-degree polynomial, the second derivative gives a second-degree polynomial, and so on. By the time we reach the fourth derivative, we are left with a constant.

Since constants have no variable components, their derivative is zero. Therefore, any derivatives of a constant, known as higher-order derivatives beyond this point, will remain zero. Consequently, the fifth derivative of the polynomial in the exercise, \(d^5y/dx^5\), is guaranteed to be zero as there are no more \(x\)-dependent terms to differentiate.