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Chapter 2: Problem 125

What is the difference between the degree and the order of a derivative?

Short Answer

Expert verified
Question: What is the difference between the degree and order of a derivative, and provide an example to illustrate the concepts. Answer: The degree of a derivative refers to the highest power of the derivative in the given function, while the order indicates how many times the function has been differentiated. For instance, given the function f(x) = x^3 - 4x^2 + 2x - 1, the degree of the original function is 3, and its first-order derivative, f'(x) = 3x^2 - 8x + 2, has a degree of 2 with an order of 1.

Step by step solution

01

Define the Degree of a Derivative

The degree of a derivative refers to the highest power of the derivative in the given function. For example, if we have a polynomial function, the degree is determined by the highest power of the variable x present in the function.
02

Define the Order of a Derivative

The order of a derivative refers to how many times a function has been differentiated. For example, if a function has been differentiated once, it is a first-order derivative. If it has been differentiated twice, it is a second-order derivative, and so on.
03

Explain the Difference

The main difference between the degree and the order of a derivative lies in what they represent. The degree refers to the highest power of the derivative in the function, whereas the order indicates how many times the function has been differentiated.
04

Provide an Example

Let's consider a function f(x) = x^3 - 4x^2 + 2x - 1. The degree of this function is 3, as the highest power of x is 3 (x^3 term). Now, let's differentiate this function once: f'(x) = 3x^2 - 8x + 2. The degree of the first-order derivative (f'(x)) is 2, as the highest power of x is 2 (3x^2 term). The order of this derivative is 1, as it has been derived once from the original function. If we differentiate f'(x) again: f''(x) = 6x - 8. The degree of the second-order derivative (f''(x)) is 1, as the highest power of x is 1 (6x term). The order of this derivative is 2, as it has been derived twice from the original function.

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