Question

Find the requested derivative. \(f(x)=\left(x^{3}-5 x+2\right)\left(x^{2}+x-7\right) ;\) find \(f^{(8)}(x)\).

Short answer

The 8th derivative \(f^{(8)}(x)\) is 0.

Step-by-step solution

Understand the Problem

You need to find the 8th derivative, denoted as \(f^{(8)}(x)\), of the function \(f(x) = \left(x^{3} - 5x + 2\right)\left(x^{2} + x - 7\right)\). This involves using knowledge of derivatives and rules for differentiation.

Identify Derivative Properties

Recognize that the functions \(x^3 - 5x + 2\) and \(x^2 + x - 7\) are both polynomials. The degree of a polynomial function is important to determine when its derivatives become zero. A polynomial of degree \(n\) will have its \((n+1)\)-th and higher derivatives equal to 0.

Calculate Degrees of Polynomials

The first polynomial within \(f(x) = x^3 - 5x + 2\) is of degree 3, and the second polynomial \(x^2 + x - 7\) is of degree 2. Multiplying these gives \(f(x)\) as a polynomial of degree 5 because \(3 + 2 = 5\).

Assess Higher Derivatives

Since \(f(x)\) is a 5th degree polynomial, any derivative taken more than 5 times will result in 0. Therefore, the 8th derivative \(f^{(8)}(x)\) will be 0, because the 6th and all higher derivatives of a 5th degree polynomial are 0.

Key concepts

Understanding Polynomial Degree

Polynomial degree is a crucial concept when dealing with differentiation, especially in higher order derivatives. The degree of a polynomial is the highest power of the variable in the expression. For example, in the function \(x^3 - 5x + 2\), the degree is 3, as the highest power of \(x\) is 3. Similarly, for \(x^2 + x - 7\), the degree is 2. When multiplying polynomials, as seen in the function \(f(x) = (x^3 - 5x + 2)(x^2 + x - 7)\), the degrees add up. Thus, the degree of \(f(x)\) is 5, because \(3 + 2 = 5\).

Understanding polynomial degree is essential in finding when the derivatives will become zero. For any polynomial of degree \(n\), the \((n+1)\)-th derivative and all higher derivatives become zero. This property simplifies solving problems involving higher order derivatives, as once you find the degree, you know that the derivatives beyond this degree plus one will vanish.

Exploring Derivative Properties

Derivative properties help us to understand and predict the behavior of polynomial functions as we differentiate them multiple times. One key property is that as you continue taking derivatives of a polynomial function, the polynomial's degree decreases by 1 with each derivative. Eventually, this process leads to the function becoming a constant, and further differentiation results in zero.

For instance, with the polynomial \(f(x)\) of degree 5, each successive derivative reduces the degree. By the time we take the 6th derivative, we are left with differentiating a zero-degree polynomial, which is a constant. Therefore, any further derivatives, such as the 8th derivative, will also be zero. This property of becoming zero after sufficiently many derivatives is what allows mathematicians to predict the 8th derivative without direct computation.

Applying Rules for Differentiation

Differentiation is a core concept in calculus and involves applying specific rules to find derivatives of functions. These rules include the power rule, product rule, and others, which allow us to find the rate of change of functions efficiently.

In the given function \(f(x)\), to initially find derivatives, one might start by using the product rule, as \(f(x)\) is a product of two polynomials, \((x^3 - 5x + 2)\) and \((x^2 + x - 7)\). The product rule states: if \(u(x)\) and \(v(x)\) are functions, then \( (uv)' = u'v + uv'\). This rule helps in differentiating products of functions.

However, once you know the degree of the polynomial function, the task becomes simpler. After identifying \(f(x)\) as a 5th degree polynomial, you can confidently say that the 8th derivative \(f^{(8)}(x)\) is zero due to the nature of polynomial degree and its derivative properties, bypassing the need for complex calculations.