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Partial Derivatives: Unlocking Multivariable Calculus
Dive into the world of partial derivatives. Learn how to analyze complex functions, solve optimization problems, and apply these powerful tools in physics, economics, and engineering.

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  2. Examples0/7 watched
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Now Playing:Partial derivatives – Example 0a
Intros
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  1. Partial Derivatives Overview:
  2. Partial Derivatives Overview:
    Introduction to Partial Derivatives
    • Derivatives in terms of 1 variable
    • Treating all other variables as constants
    • An example
  3. Partial Derivatives Overview:
    Definition of Partial Derivatives
    • Recalling the definition of derivative
    • Two formal equations
    • Won't be Using them (Too Complicated)
Examples
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  1. Finding the Partial Derivatives
    Find the first order partial derivatives of the following function:

    f(x,y)=2xln⁡(xy2)+xy−x3f(x,y) = 2x \ln (xy^2) + \frac{x}{y} - \sqrt{x^3}

    Functions of several variables
    Jump to:NotesConceptFAQsPrerequisites
    Notes
    Notes:

    Introduction to Partial Derivatives
    Since we are dealing with multi-variable functions, we want to take the derivative in respect to 1 variable. This is known as the partial derivative.

    For a function f(x,y)f(x,y), we can have two partial derivatives:
    • fx=dfdx→f_x = \frac{df}{dx} \to derivative in terms of xx
    • fy=dfdy→f_y = \frac{df}{dy} \to derivative in terms of yy
    When taking partial derivatives of functions in terms of a variable, we treat other variables as constants. In other words, we allow one variable to vary, and the other variables to be fixed.

    Definition of Partial Derivatives
    Recall from the definition of derivative, we have the formula:

    lim⁡h→0f(x+h)−f(x)h\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}

    From for the definition of partial derivatives, we have the two following equations:

    fx=lim⁡h→0f(x+h,y)−f(x,y)h f_x = \lim\limits_{h \to 0} \frac{f(x+h,y) - f(x,y)}{h}
    fy=lim⁡h→0f(x,y+h)−f(x,y)h f_y = \lim\limits_{h \to 0} \frac{f(x, y+h) - f(x,y)}{h}

    We won't be using these equations because we already know how to take derivatives.

    Application of Partial Derivatives
    We can use partial derivatives to find the tangent slope of the traces at a certain point (a,b)(a,b). We can do this by finding fx(a,b)f_x(a,b) and fy(a,b)f_y(a,b).
    We can also use partial derivatives to see if f(x,y)f(x,y) is increasing or decreasing. In other words, if

    fx>0f_x\gt0, then f(x,y)f(x,y) is increasing as we vary xx.
    fx<0f_x\lt0, then f(x,y)f(x,y) is decreasing as we vary xx.
    fy>0f_y\gt0, then f(x,y)f(x,y) is increasing as we vary yy.
    fy<0f_y\lt0, then f(x,y)f(x,y) is decreasing as we vary yy.
    Concept

    Introduction to Partial Derivatives

    Partial derivatives are a fundamental concept in calculus, particularly when dealing with multivariable functions. They allow us to analyze how a function changes with respect to one variable while holding others constant. This powerful tool is essential in various fields, including physics, economics, and engineering. Partial derivatives extend the idea of single-variable derivatives to functions of multiple variables, enabling us to explore complex relationships and optimize multidimensional systems. The introduction video provides a clear and concise explanation of partial derivatives, making it an invaluable resource for students and professionals alike. By watching this video, viewers will gain a solid foundation in understanding how partial derivatives work, their notation, and their applications in real-world scenarios. Mastering partial derivatives is crucial for anyone studying advanced mathematics or working with multivariable functions, as they form the basis for many higher-level concepts in calculus and mathematical analysis.

    FAQs

    1. What is a partial derivative?

      A partial derivative measures how a function changes with respect to one variable while holding all other variables constant. It's denoted by f/x for a function f with respect to variable x.

    2. How do you calculate a partial derivative?

      To calculate a partial derivative, treat all variables except the one you're differentiating with respect to as constants. Then, apply standard differentiation rules to the variable of interest.

    3. What's the difference between a partial derivative and a regular derivative?

      A regular derivative deals with functions of one variable, while partial derivatives are used for functions of multiple variables. Partial derivatives focus on how a function changes with respect to one variable while keeping others constant.

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

      Partial derivatives are used in physics to analyze heat flow and fluid dynamics, in economics to study marginal costs and benefits, and in engineering for optimization problems. They're also crucial in machine learning algorithms like gradient descent.

    5. How do partial derivatives extend to higher dimensions?

      In higher dimensions, the concept remains the same, but you have more variables to consider. For a function with n variables, you can take n different partial derivatives, each with respect to one of the variables while treating the others as constants.

    Prerequisites

    Mastering partial derivatives requires a solid foundation in several key mathematical concepts. One of the most fundamental prerequisites is understanding the rate of change. This concept is crucial because partial derivatives essentially measure the rate at which a function changes with respect to one variable while holding others constant.

    Building on this, students must be proficient in determining the slope and equation of tangent line. This skill is directly applicable to partial derivatives, as they represent the slope of the tangent plane to a surface at a specific point. Understanding how to calculate and interpret these slopes in two dimensions prepares students for the more complex three-dimensional scenarios encountered in partial derivatives.

    Another critical prerequisite is the chain rule. While the chain rule is initially learned for single-variable calculus, its principles extend to partial derivatives when dealing with composite functions of multiple variables. Grasping this concept is essential for tackling more advanced problems in multivariable calculus.

    Additionally, familiarity with the power of a product rule lays the groundwork for understanding how partial derivatives behave with respect to products of functions. This algebraic principle is frequently applied when differentiating complex multivariable functions.

    Each of these prerequisite topics contributes uniquely to the understanding of partial derivatives. The rate of change concept provides the fundamental idea of how quantities vary, which is then extended to multiple dimensions in partial derivatives. The slope of tangent lines directly relates to the geometric interpretation of partial derivatives as slopes of planes tangent to surfaces.

    The chain rule becomes indispensable when dealing with composite functions in multiple variables, allowing students to break down complex expressions into manageable parts. Lastly, the power of a product rule aids in handling the intricate algebraic manipulations often required when working with partial derivatives of multivariable functions.

    By thoroughly understanding these prerequisite topics, students can approach partial derivatives with a strong conceptual and practical foundation. This preparation not only makes learning partial derivatives more accessible but also enhances the ability to apply them in various fields such as physics, engineering, and economics, where multivariable calculus plays a crucial role.