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Higher Order Partial Derivatives: Advanced Techniques and Applications
Dive deep into higher order partial derivatives. Learn advanced calculus concepts, problem-solving strategies, and real-world applications in physics, engineering, and economics.

  1. Intros0/3 watched
  2. Examples0/6 watched
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Now Playing:Higher order partial derivatives – Example 0a
Intros
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  1. High Order Partial Derivatives Overview:
  2. High Order Partial Derivatives Overview:
    2nd Order Partial Derivatives
    • 4 types of 2nd order partial derivatives
    • fxx,fxy,fyy,fyxf_{xx}, f_{xy}, f_{yy}, f_{yx}
    • An example
  3. High Order Partial Derivatives Overview:
    Higher Order Partial Derivatives
    • Can go higher than 2nd order
    • fxxx,fxxy,fxxxxxf_{xxx}, f_{xxy}, f_{xxxxx}
    • An example
Examples
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  1. Finding 2nd Order Partial Derivatives
    Find all the second order derivatives for the following function

    f(x,y)=x3y−4xy3+ln⁡(x2) f(x,y) = x^3 y - \sqrt{4xy^3} + \ln (x^2)

    Functions of several variables
    Jump to:NotesConceptExampleFAQsPrerequisites
    Notes
    Notes:

    2nd Order Partial Derivatives
    Since we can have higher order derivatives on a one variable function, we can also have this for multi-variable functions. We will specifically look at 2nd order partial derivatives here. For 2nd order partial derivatives, there are 4 types:

    fxx=ddx(dfdx)=d2fdx2f_{xx} = \frac{d}{dx}(\frac{df}{dx}) = \frac{d^2f}{dx^2}
    fxy=ddy(dfdx)=d2fdydxf_{xy} = \frac{d}{dy}(\frac{df}{dx}) = \frac{d^2f}{dydx}
    fyy=ddy(dfdy)=d2fdy2f_{yy} = \frac{d}{dy}(\frac{df}{dy}) = \frac{d^2f}{dy^2}
    fyx=ddx(dfdy)=d2fdxdyf_{yx} = \frac{d}{dx}(\frac{df}{dy}) = \frac{d^2f}{dxdy}

    Where:
    fxx→f_{xx} \to derivative in respect to xx 2 times
    fyy→f_{yy} \to derivative in respect to yy 2 times
    fxy→f_{xy} \to derivative in respect to xx first, and then respect to yy
    fyx→f_{yx} \to derivative in respect to yy first, and then respect to xx

    Higher Order Partial Derivatives
    Of course, we can have even higher order partial derivatives. For example, we can have:

    fxxx=ddx(d2fdx2)=d3fdx3f_{xxx} = \frac{d}{dx} (\frac{d^2f}{dx^2}) = \frac{d^3f}{dx^3}
    fxxy=ddy(d2fdx2)=d3fdydx2f_{xxy} = \frac{d}{dy} (\frac{d^2f}{dx^2}) = \frac{d^3f}{dydx^2}
    fxxxxx=ddx(d4fdx4)=d5fdx5f_{xxxxx} = \frac{d}{dx} (\frac{d^4f}{dx^4}) = \frac{d^5f}{dx^5}

    We cannot list them all here because there is an infinite amount of higher order partial derivatives.

    Clairaut's Theorem Suppose that ff is defined on a disk DD that contains the point (a,b)(a,b). If the functions fxyf_{xy} and fyxf_{yx} are continuous on this disk, then

    fxy(a,b)=fyx(a,b)f_{xy}(a,b) = f_{yx}(a,b)

    Concept

    Introduction to Higher Order Partial Derivatives

    Higher order partial derivatives are an advanced concept in multivariable calculus that builds upon the foundation of partial derivatives. The introduction video serves as a crucial starting point for understanding this complex topic. It demonstrates how to extend the idea of partial derivatives to higher orders, allowing for a more in-depth analysis of multivariable functions. By grasping higher order partial derivatives, students can explore the curvature and behavior of surfaces in multiple dimensions. This concept is essential for various applications in physics, engineering, and economics. The video explains how to calculate these derivatives systematically, emphasizing the importance of notation and order of differentiation. As students progress from basic partial derivatives to higher orders, they gain a more comprehensive understanding of function behavior and can tackle more sophisticated problems in mathematical modeling and optimization. This knowledge is fundamental for advanced studies in calculus and its real-world applications.

    Example

    High Order Partial Derivatives Overview: 2nd Order Partial Derivatives

    • 4 types of 2nd order partial derivatives
    • fxx,fxy,fyy,fyxf_{xx}, f_{xy}, f_{yy}, f_{yx}
    • An example

    Step 1: Introduction to Second-Order Partial Derivatives

    In this section, we will explore second-order partial derivatives. Recall from Calculus 1 that taking the derivative of a function and then taking the derivative again results in a second-order derivative. Similarly, in the context of partial derivatives, we can take second-order partial derivatives. However, since we are dealing with functions of multiple variables, there are more types of second-order partial derivatives to consider.

    Step 2: Types of Second-Order Partial Derivatives

    There are four types of second-order partial derivatives when dealing with functions of two variables, xx and yy:

    • fxxf_{xx}: The second partial derivative with respect to xx twice.
    • fxyf_{xy}: The partial derivative with respect to xx first, and then with respect to yy.
    • fyyf_{yy}: The second partial derivative with respect to yy twice.
    • fyxf_{yx}: The partial derivative with respect to yy first, and then with respect to xx.

    Step 3: Example Function

    Consider the function f(x,y)=x2y+y2xf(x, y) = x^2 y + y^2 x. We will find the second-order partial derivative fxxf_{xx}.

    Step 4: First Partial Derivative with Respect to xx

    To find fxxf_{xx}, we first need to find the first partial derivative of ff with respect to xx. Treat yy as a constant:

    • The derivative of x2yx^2 y with respect to xx is 2xy2xy.
    • The derivative of y2xy^2 x with respect to xx is y2y^2.
    Therefore, the first partial derivative fxf_x is fx=2xy+y2f_x = 2xy + y^2.

    Step 5: Second Partial Derivative with Respect to xx

    Next, we take the partial derivative of fxf_x with respect to xx again:

    • The derivative of 2xy2xy with respect to xx is 2y2y.
    • The derivative of y2y^2 with respect to xx is 00 since y2y^2 is a constant with respect to xx.
    Therefore, the second partial derivative fxxf_{xx} is fxx=2yf_{xx} = 2y.

    Step 6: Summary

    We have found that for the function f(x,y)=x2y+y2xf(x, y) = x^2 y + y^2 x, the second-order partial derivative fxxf_{xx} is 2y2y. This process can be repeated for the other second-order partial derivatives fxyf_{xy}, fyyf_{yy}, and fyxf_{yx}, which will be covered in subsequent examples.

    FAQs

    1. What are higher order partial derivatives?

      Higher order partial derivatives are derivatives of a multivariable function taken multiple times with respect to one or more variables. They extend beyond first-order partial derivatives and provide deeper insights into the behavior of functions. For example, a second-order partial derivative might be taken twice with respect to x (fxx), or once with respect to x and once with respect to y (fxy).

    2. How do you calculate a second-order partial derivative?

      To calculate a second-order partial derivative, you take the partial derivative of the function twice. For example, to find fxy of f(x,y) = x²y + y²x: 1. First, find fx: fx = 2xy + y² 2. Then, differentiate fx with respect to y: fxy = 2x + 2y

    3. What is Clairaut's Theorem and why is it important?

      Clairaut's Theorem states that for a function with continuous second partial derivatives, the order of differentiation doesn't matter for mixed partial derivatives (e.g., fxy = fyx). This theorem is important because it simplifies calculations and verifies the consistency of mathematical models in various fields like physics and engineering.

    4. What are some real-world applications of higher order partial derivatives?

      Higher order partial derivatives have numerous applications, including: - In physics for describing acceleration and wave equations - In economics for analyzing elasticity and optimizing economic models - In engineering for structural analysis and control systems - In computer graphics for edge detection and image processing These derivatives help model complex systems and phenomena across various disciplines.

    5. What are some common challenges in solving higher order partial derivative problems?

      Common challenges include: - Understanding complex notation - Keeping track of which variables are held constant - Applying the chain rule correctly - Interpreting the geometric meaning of higher order derivatives - Relating abstract concepts to real-world applications Overcoming these challenges requires practice, systematic problem-solving approaches, and relating mathematical concepts to physical phenomena.

    Prerequisites

    Understanding higher order partial derivatives is a crucial concept in advanced calculus and multivariable analysis. To fully grasp this topic, it's essential to have a solid foundation in several prerequisite areas. One of the fundamental concepts you should be familiar with is the power of a product rule. This algebraic principle is vital when dealing with complex functions involving multiple variables and their derivatives.

    As you delve deeper into calculus, the chain rule becomes an indispensable tool. This rule is particularly relevant to higher order partial derivatives as it allows you to differentiate composite functions, which are common in multivariable calculus. Mastering the chain rule will significantly enhance your ability to compute complex partial derivatives efficiently.

    Another important prerequisite topic is Taylor series and Maclaurin series. These series expansions play a crucial role in approximating functions and are often used in conjunction with higher order partial derivatives. Understanding Taylor series will help you analyze the behavior of multivariable functions and their derivatives in more detail.

    Lastly, familiarity with exact differential equations is highly beneficial when studying higher order partial derivatives. This topic provides insights into the relationships between partial derivatives and helps in solving certain types of differential equations that arise in multivariable calculus.

    By mastering these prerequisite topics, you'll build a strong foundation for understanding higher order partial derivatives. The power of a product rule will aid in manipulating complex expressions, while the chain rule will be essential for differentiating composite functions. Taylor series will provide tools for approximation and analysis, and knowledge of exact differential equations will deepen your understanding of the relationships between partial derivatives.

    As you progress in your study of higher order partial derivatives, you'll find that these prerequisite topics are not just isolated concepts but interconnected tools that work together. For instance, you might use the chain rule in conjunction with the power of a product rule when differentiating complex multivariable functions. Similarly, Taylor series expansions often involve higher order partial derivatives, demonstrating the cyclical nature of these mathematical concepts.

    In conclusion, a thorough understanding of these prerequisite topics will not only make learning higher order partial derivatives easier but also provide you with a more comprehensive view of multivariable calculus. By building on this strong foundation, you'll be better equipped to tackle advanced problems and applications in fields such as physics, engineering, and economics where higher order partial derivatives are frequently encountered.