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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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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Degree of a Derivative
In calculus, understanding the degree of a derivative helps clarify the behavior of functions, especially polynomials, when differentiated. The degree is determined by the highest power of the variable in the derivative itself. In simpler terms, it's the largest exponent seen in the expression after taking the derivative.
  • Consider a polynomial like \( f(x) = x^3 - 4x^2 + 2x - 1 \). Here, before differentiation, the degree is 3, owing to the \( x^3 \) term.
  • Upon differentiating the function once, we get \( f'(x) = 3x^2 - 8x + 2 \). The degree drops to 2 due to the \( 3x^2 \) term.
  • Continuing to the second derivative, \( f''(x) = 6x - 8 \), the degree becomes 1, reflecting the \( 6x \) term.

This change in degree highlights how each differentiation step tends to reduce the complexity (or the degree) of the polynomial, bringing it closer to a linear function.
Order of a Derivative
The order of a derivative is a fundamental concept in calculus that provides insight into the number of times a function has been differentiated. Each time you differentiate, the order increases by one.
For example, starting with a function \( g(x) \):
  • The original function \( g(x) \) is naturally a zero-order derivative as it hasn't been differentiated yet.
  • If differentiated once, the resulting function, \( g'(x) \), becomes the first-order derivative.
  • Another differentiation gives you \( g''(x) \), a second-order derivative.
It is important to note that the order of a derivative tells you nothing about the degree of the function, only how many times differentiation has been applied. This is particularly useful in fields like physics and engineering, where higher-order derivatives relate to concepts like acceleration and jerk.
Differentiation in Calculus
Differentiation is a core operation in calculus, essential for studying changes and trends in functions. It involves calculating derivatives to understand how a quantity changes relative to change in one of its parameters. This is pivotal for comprehending motion, rates of change, and optimizing solutions.
  • Differentiation allows us to find the slope of a curve at any point, providing insights into the rate of change.
  • This process not only identifies how fast things are changing but also the nature of that change, such as increasing or decreasing trends.
  • Beyond such basic calculations, differentiation is crucial in real-world applications like evaluating the marginal cost in economics or determining instantaneous velocity in physics.

Through understanding and applying differentiation, students can analyze functions more thoroughly, predicting and interpreting behaviors across myriad disciplines. Mastery of differentiation, along with knowledge of the degree and order of derivatives, opens doors to deeper insights into both theoretical and applied mathematics.

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Most popular questions from this chapter

Consider a large plane wall of thickness \(L=0.3 \mathrm{~m}\), thermal conductivity \(k=2.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and surface area \(A=\) \(12 \mathrm{~m}^{2}\). The left side of the wall at \(x=0\) is subjected to a net heat flux of \(\dot{q}_{0}=700 \mathrm{~W} / \mathrm{m}^{2}\) while the temperature at that surface is measured to be \(T_{1}=80^{\circ} \mathrm{C}\). Assuming constant thermal conductivity and no heat generation in the wall, \((a)\) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the wall, \((b)\) obtain a relation for the variation of temperature in the wall by solving the differential equation, and \((c)\) evaluate the temperature of the right surface of the wall at \(x=L\).

The temperatures at the inner and outer surfaces of a 15 -cm-thick plane wall are measured to be \(40^{\circ} \mathrm{C}\) and \(28^{\circ} \mathrm{C}\), respectively. The expression for steady, one-dimensional variation of temperature in the wall is (a) \(T(x)=28 x+40\) (b) \(T(x)=-40 x+28\) (c) \(T(x)=40 x+28\) (d) \(T(x)=-80 x+40\) (e) \(T(x)=40 x-80\)

Consider a small hot metal object of mass \(m\) and specific heat \(c\) that is initially at a temperature of \(T_{i}\). Now the object is allowed to cool in an environment at \(T_{\infty}\) by convection with a heat transfer coefficient of \(h\). The temperature of the metal object is observed to vary uniformly with time during cooling. Writing an energy balance on the entire metal object, derive the differential equation that describes the variation of temperature of the ball with time, \(T(t)\). Assume constant thermal conductivity and no heat generation in the object. Do not solve.

Consider steady one-dimensional heat conduction in a plane wall in which the thermal conductivity varies linearly. The error involved in heat transfer calculations by assuming constant thermal conductivity at the average temperature is \((a)\) none, \((b)\) small, or \((c)\) significant.

Heat is generated in a \(3-\mathrm{cm}\)-diameter spherical radioactive material uniformly at a rate of \(15 \mathrm{~W} / \mathrm{cm}^{3}\). Heat is dissipated to the surrounding medium at \(25^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(120 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The surface temperature of the material in steady operation is (a) \(56^{\circ} \mathrm{C}\) (b) \(84^{\circ} \mathrm{C}\) (c) \(494^{\circ} \mathrm{C}\) (d) \(650^{\circ} \mathrm{C}\) (e) \(108^{\circ} \mathrm{C}\)
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