Glossary

3.2 Higher-order partial derivatives

3 min read•august 6, 2024

Higher-order partial derivatives take calculus to the next level. They let us analyze functions of multiple variables more deeply, revealing how changes in one variable affect another's rate of change.

These derivatives are crucial for understanding complex systems. They help us find , approximate functions, and study their behavior. Mastering them opens doors to advanced math and real-world applications.

Second-Order Partial Derivatives

Pure and Mixed Partial Derivatives

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  • are obtained by taking the partial derivative of a partial derivative
  • involve taking the partial derivative of a function with respect to the same variable twice
    • For a function f(x,y)f(x, y), the pure partial derivatives are denoted as ∂2f∂x2\frac{\partial^2 f}{\partial x^2} and ∂2f∂y2\frac{\partial^2 f}{\partial y^2}
    • Example: If f(x,y)=x2y+xy2f(x, y) = x^2y + xy^2, then ∂2f∂x2=2y\frac{\partial^2 f}{\partial x^2} = 2y and ∂2f∂y2=2x\frac{\partial^2 f}{\partial y^2} = 2x
  • involve taking the partial derivative of a function with respect to one variable and then taking the partial derivative of the result with respect to another variable
    • For a function f(x,y)f(x, y), the mixed partial derivatives are denoted as ∂2f∂x∂y\frac{\partial^2 f}{\partial x \partial y} and ∂2f∂y∂x\frac{\partial^2 f}{\partial y \partial x}
    • Example: If f(x,y)=x2y+xy2f(x, y) = x^2y + xy^2, then ∂2f∂x∂y=2x+2y\frac{\partial^2 f}{\partial x \partial y} = 2x + 2y and ∂2f∂y∂x=2x+2y\frac{\partial^2 f}{\partial y \partial x} = 2x + 2y

Advanced Topics in Higher-Order Partial Derivatives

Hessian Matrix

  • The is a square matrix of second-order partial derivatives for a function of multiple variables
  • For a function f(x1,x2,…,xn)f(x_1, x_2, \ldots, x_n), the Hessian matrix is defined as: \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1 \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1 \partial x_n} \\ \frac{\partial^2 f}{\partial x_2 \partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2 \partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2 f}{\partial x_n \partial x_1} & \frac{\partial^2 f}{\partial x_n \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{bmatrix}$$
  • The Hessian matrix is used to analyze the local behavior of a function, such as determining the type of critical points (minimum, maximum, or saddle points)

Taylor Series Expansion

  • is a method for approximating a function near a point using a polynomial series
  • For a function f(x,y)f(x, y) with continuous partial derivatives up to order nn, the Taylor series expansion around the point (a,b)(a, b) is given by: f(x,y)≈∑i=0n∑j=0n−i1i!j!∂i+jf∂xi∂yj(a,b)(x−a)i(y−b)jf(x, y) \approx \sum_{i=0}^n \sum_{j=0}^{n-i} \frac{1}{i!j!} \frac{\partial^{i+j} f}{\partial x^i \partial y^j}(a, b)(x-a)^i(y-b)^j
  • The Taylor series expansion is useful for approximating functions and analyzing their behavior near a specific point

Schwarz's Theorem

  • , also known as the or the equality of mixed partials, states that under certain conditions, the order of taking mixed partial derivatives does not matter
  • If the mixed partial derivatives ∂2f∂x∂y\frac{\partial^2 f}{\partial x \partial y} and ∂2f∂y∂x\frac{\partial^2 f}{\partial y \partial x} are continuous on an open set containing the point (a,b)(a, b), then: ∂2f∂x∂y(a,b)=∂2f∂y∂x(a,b)\frac{\partial^2 f}{\partial x \partial y}(a, b) = \frac{\partial^2 f}{\partial y \partial x}(a, b)
  • Schwarz's theorem simplifies the computation of higher-order partial derivatives and ensures the existence of a unique Hessian matrix for a function satisfying the conditions

Key Terms to Review (19)

∂²f/∂x∂y: The expression ∂²f/∂x∂y represents the mixed second-order partial derivative of a function f with respect to two variables, x and y. It is calculated by first taking the partial derivative of the function f with respect to y and then taking the partial derivative of that result with respect to x. This operation is crucial for understanding how the function behaves in relation to changes in both variables, and it plays a significant role in various applications, such as optimization and analyzing the curvature of surfaces.
∂²f/∂x²: The term $$\frac{\partial^2 f}{\partial x^2}$$ represents the second-order partial derivative of a function $$f$$ with respect to the variable $$x$$. It measures how the rate of change of the function $$f$$ with respect to $$x$$ changes as $$x$$ itself changes, essentially providing insights into the curvature or concavity of the function along that axis. This concept is crucial in understanding how functions behave in multivariable calculus, particularly in optimization problems and differential equations.
∂²f/∂y∂x: The term ∂²f/∂y∂x represents a mixed second-order partial derivative of a function f with respect to the variables x and y. This derivative measures how the rate of change of the function f changes as both variables vary, first with respect to x and then with respect to y. Understanding this concept is crucial for analyzing how functions behave in multiple dimensions, as it reveals information about the function's curvature and interaction between variables.
∂²f/∂y²: The notation ∂²f/∂y² represents the second partial derivative of a function $$f$$ with respect to the variable $$y$$. This term measures how the rate of change of the function $$f$$ with respect to $$y$$ itself changes as $$y$$ varies, providing insight into the curvature and concavity of the function in a multi-variable context. Understanding second partial derivatives is crucial for analyzing the behavior of functions in higher dimensions, especially when determining local maxima, minima, and points of inflection.
Clairaut's Theorem: Clairaut's Theorem states that if a function has continuous second partial derivatives, then the order of differentiation does not matter; that is, the mixed partial derivatives are equal. This theorem is important in understanding how to compute higher-order derivatives and ensures that we can interchange the order of differentiation for functions with certain smoothness conditions, linking it to both the definition of partial derivatives and higher-order partial derivatives.
Concavity: Concavity refers to the direction in which a curve bends. A function is concave up if it curves upwards, resembling a cup, while it is concave down if it curves downwards, resembling an arch. The concept of concavity is crucial for understanding the behavior of functions, particularly in analyzing their curvature, which is closely tied to second derivatives and the geometric interpretation of graphs.
Continuity: Continuity is a property of functions that describes the behavior of a function at a point, ensuring that small changes in input result in small changes in output. It is crucial for understanding how functions behave, particularly when dealing with limits, derivatives, and integrals across multiple dimensions.
Convexity: Convexity refers to the property of a set or a function where any line segment drawn between two points within that set or function lies entirely within the set or above the function's graph. In relation to curves and surfaces, a convex shape curves outward, which affects properties like arc length and curvature, as well as how functions behave under higher-order derivatives. Understanding convexity is crucial for analyzing geometric features and optimizing functions in various contexts.
Critical Points: Critical points are locations in a function where the derivative is either zero or undefined, indicating potential local maxima, minima, or saddle points. Understanding critical points is essential as they play a crucial role in analyzing the behavior of functions, optimizing values, and determining overall trends in higher dimensions.
Differentiability: Differentiability refers to the property of a function where it has a derivative at a given point, meaning the function can be locally approximated by a linear function. This concept is essential for understanding how functions behave near specific points, allowing us to analyze and predict their behavior in various contexts, including surfaces, extrema, and integrals.
Hessian Matrix: The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function. It provides crucial information about the curvature of the function at a given point and is particularly important in optimization problems, where it helps identify local maxima, minima, or saddle points of functions with multiple variables.
Mixed Partial Derivatives: Mixed partial derivatives refer to the second-order derivatives of a multivariable function, where the differentiation is performed with respect to different variables in succession. This concept helps analyze how a function behaves with respect to changes in multiple inputs, revealing interactions between those variables. Mixed partial derivatives are crucial when dealing with functions of two or more variables, especially in optimization problems and understanding surface curvature.
Partial Differentiation: Partial differentiation is a technique used in calculus to find the derivative of a multivariable function with respect to one variable while keeping other variables constant. This concept is essential for understanding how functions behave when they depend on multiple variables, allowing for analysis of their local behavior and optimization. Higher-order partial derivatives extend this idea by considering the derivatives of the first partial derivatives, providing deeper insight into the curvature and behavior of multivariable functions.
Pure Partial Derivatives: Pure partial derivatives refer to the derivatives of a function with respect to one variable while holding all other variables constant. This concept is crucial for analyzing functions of multiple variables, allowing us to understand how changes in a single variable affect the function's value without interference from other variables.
Schwarz's Theorem: Schwarz's Theorem, also known as Schwarz's symmetry theorem, states that for a function that is continuously differentiable and has continuous second partial derivatives, the order of differentiation does not matter. This means that if you take the partial derivatives of a function in two different orders, the results will be the same. This theorem is crucial for higher-order partial derivatives as it guarantees that mixed partial derivatives are equal under these conditions.
Second-order partial derivatives: Second-order partial derivatives are the derivatives of first-order partial derivatives, which measure how a multivariable function changes as one variable changes while keeping other variables constant. They provide insight into the curvature and behavior of functions in higher dimensions, allowing for an analysis of local maxima, minima, and saddle points. Understanding these derivatives is crucial for optimizing functions and analyzing their properties.
Symmetry of Second Derivatives: The symmetry of second derivatives refers to the property that, for a function with continuous second partial derivatives, the mixed partial derivatives are equal. This means that if you take the second partial derivative of a function first with respect to one variable and then with respect to another, it will yield the same result as if you reversed the order of differentiation. This property is essential when dealing with higher-order partial derivatives, ensuring consistent results in calculations.
Taylor Series Expansion: A Taylor series expansion is an infinite sum of terms calculated from the values of a function's derivatives at a single point. This mathematical concept helps to approximate complex functions using polynomials, making it easier to analyze and compute values for functions that may be difficult to work with directly. It connects to understanding higher-order derivatives and can also be applied in situations involving implicit differentiation through the chain rule, which is essential for understanding how functions behave locally around specific points.
Total Differentiation: Total differentiation is the process of computing the differential of a function that depends on multiple variables, capturing how changes in those variables collectively affect the function's value. This concept not only incorporates the partial derivatives of the function but also considers how each variable interacts with one another, providing a comprehensive view of how the function behaves under small changes. Understanding total differentiation is crucial for applying higher-order partial derivatives, as it helps in exploring the behavior of multivariable functions and their rates of change.
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