Problem 18
Question
Find a formula for $$ D_{x}^{n}\left(a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}\right) $$
Step-by-Step Solution
Verified Answer
The \(n\)-th derivative of the polynomial is \(0\).
1Step 1: Understand the Problem
The exercise requires finding a formula for the derivative of a polynomial of degree \(n-1\) taken \(n\) times. That's equivalent to differentiating an \((n-1)\) degree polynomial \(n\) times with respect to \(x\).
2Step 2: Write Down the General Polynomial
The polynomial is given as \(a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \, ... \, + a_{1} x + a_{0}\), where each term is of the form \(a_k x^k\) and \(k\) ranges from \(0\) to \(n-1\).
3Step 3: Apply the Power Rule
According to the power rule, the derivative of \(a_k x^k\) with respect to \(x\) is \(k \, a_k x^{k-1}\). The repetitive application of this rule \(n\) times is required for each of the terms in the polynomial.
4Step 4: Identify the Effect of Differentiation
During each differentiation, the power of \(x\) in each term decreases by one. After differentiating \(n\) times, any term \(a_k x^k\) with \(k
5Step 5: Analyze the Remaining Terms
As the polynomial is of degree \(n-1\), all terms will vanish when derivative is taken \(n\) times. This is because a \(k\) degree polynomial can only be differentiated \(k\) times before it becomes zero.
6Step 6: Find the Final Result
Since all terms in the polynomial will have been differentiated away by the \(n\)-th differentiation, the result of \(D_x^n(a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0})\) is \(0\).
Key Concepts
Polynomial DifferentiationPower RuleCalculus Concepts
Polynomial Differentiation
Polynomial differentiation is a vital concept in calculus that involves finding the derivative of a polynomial function. To grasp this fully, consider a polynomial as a function that is a sum of terms, each containing a constant coefficient and a variable raised to a positive integer power. In the given exercise, you have a polynomial of degree \(n-1\). This means its highest-powered term is \(x^{n-1}\).
When you differentiate a polynomial, you are applying the process of finding a derivative to each term individually. The derivative, in simple terms, measures how a function's output value changes as its input value changes. For a polynomial, this involves a key tool known as the power rule, which we'll explore soon.
By applying polynomial differentiation, you can find each term's rate of change and, consequently, the rate of change of the entire polynomial. In consecutive derivatives, once you exceed the degree of the polynomial, it vanishes. In our case, differentiating an \(n-1\) degree polynomial \(n\) times results in a polynomial that becomes 0, as all terms diminish completely.
When you differentiate a polynomial, you are applying the process of finding a derivative to each term individually. The derivative, in simple terms, measures how a function's output value changes as its input value changes. For a polynomial, this involves a key tool known as the power rule, which we'll explore soon.
By applying polynomial differentiation, you can find each term's rate of change and, consequently, the rate of change of the entire polynomial. In consecutive derivatives, once you exceed the degree of the polynomial, it vanishes. In our case, differentiating an \(n-1\) degree polynomial \(n\) times results in a polynomial that becomes 0, as all terms diminish completely.
Power Rule
The power rule is a fundamental technique in calculus that makes differentiating polynomials straightforward and efficient. It states that if you have a function of the form \(f(x) = x^k\), the derivative \(f'(x)\) is found by multiplying the power \(k\) by the coefficient and then reducing the power by one: \(f'(x) = kx^{k-1}\).
In the exercise, each term of the polynomial can be differentiated using the power rule. For instance, if you have a term \(a_kx^k\), applying the power rule gives you \(ka_kx^{k-1}\) as the derivative. Each differentiation lowers the power of \(x\) by one, and you multiply by the new power.
It's crucial to realize that repeated application of this rule is needed when differentiating polynomials multiple times, as in this exercise. As you differentiate, terms eventually vanish when their power becomes zero or negative, particularly after exceeding the highest degree of the original polynomial.
In the exercise, each term of the polynomial can be differentiated using the power rule. For instance, if you have a term \(a_kx^k\), applying the power rule gives you \(ka_kx^{k-1}\) as the derivative. Each differentiation lowers the power of \(x\) by one, and you multiply by the new power.
It's crucial to realize that repeated application of this rule is needed when differentiating polynomials multiple times, as in this exercise. As you differentiate, terms eventually vanish when their power becomes zero or negative, particularly after exceeding the highest degree of the original polynomial.
Calculus Concepts
Understanding calculus concepts like differentiation is essential to solving problems involving rates of change and motion, among others. Differentiation is one of the two main operations in calculus, the other being integration. It allows us to compute the derivative, which represents the rate of change of a function relative to changes in its variable.
Higher-order derivatives, as seen in the exercise, involve taking the derivative of the derivative, potentially multiple times. Each successive derivative provides further insight into the original function's behavior, such as acceleration in the context of motion.
In the case of polynomials, each differentiation reduces the degree until the polynomial becomes a constant and eventually zero. This concept ties into the understanding that polynomials, when fully differentiated, reach a stable state of zero, illustrating a key property of these seemingly simple yet profound mathematical functions. Understanding these concepts is foundational for more advanced calculus studies.
Higher-order derivatives, as seen in the exercise, involve taking the derivative of the derivative, potentially multiple times. Each successive derivative provides further insight into the original function's behavior, such as acceleration in the context of motion.
In the case of polynomials, each differentiation reduces the degree until the polynomial becomes a constant and eventually zero. This concept ties into the understanding that polynomials, when fully differentiated, reach a stable state of zero, illustrating a key property of these seemingly simple yet profound mathematical functions. Understanding these concepts is foundational for more advanced calculus studies.
Other exercises in this chapter
Problem 17
Chris, who is 6 feet tall, is walking away from a street light pole 30 feet high at a rate of 2 feet per second. (a) How fast is his shadow increasing in length
View solution Problem 18
In Problems 1-18, find \(D_{x} y\). $$ y=\sec ^{3} x $$
View solution Problem 18
Find the indicated derivative. \(D_{x} \ln \sqrt{3 x-2}\)
View solution Problem 18
Find \(D_{x} y\). $$ y=\operatorname{coth}^{-1}\left(x^{5}\right) $$
View solution