Glossary

3.3 Partial Derivatives and the Gradient

2 min read•july 25, 2024

Partial derivatives are key tools for understanding how multivariable functions change. They measure the rate of change with respect to one variable while keeping others constant, helping us analyze complex relationships in physics, economics, and more.

The combines partial derivatives, pointing in the direction of steepest ascent. It's crucial for optimization problems and understanding the geometry of multivariable functions. Higher-order derivatives provide deeper insights into function behavior and curvature.

Partial Derivatives

Partial derivatives of multivariable functions

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  • Measures rate of change of function with respect to one variable while holding others constant
  • Notation: ∂f∂x\frac{\partial f}{\partial x}, ∂f∂y\frac{\partial f}{\partial y}, [fx](https://www.fiveableKeyTerm:fx)[f_x](https://www.fiveableKeyTerm:f_x), [fy](https://www.fiveableKeyTerm:fy)[f_y](https://www.fiveableKeyTerm:f_y) represents partial derivatives
  • Calculation steps:
    1. Treat other variables as constants
    2. Apply single-variable differentiation rules
  • Common functions and their partial derivatives include polynomials (x2y→2xyx^2y \rightarrow 2xy), trigonometric functions (sin⁡(xy)→ycos⁡(xy)\sin(xy) \rightarrow y\cos(xy)), exponential and logarithmic functions (ex+y→ex+ye^{x+y} \rightarrow e^{x+y})
  • extends to partial derivatives for composite functions
  • applies to equations defining multivariable functions implicitly

Interpretation of partial derivatives

  • Represents slope of tangent line in direction of each variable
  • Relates to directional derivatives providing rates of change in specific directions
  • Applications span physics (velocity components in 3D motion) and economics (marginal cost and revenue analysis)
  • Enables sensitivity analysis assessing impact of small changes in variables
  • Approximates small changes using linear approximation formula Δf≈fxΔx+fyΔy\Delta f \approx f_x \Delta x + f_y \Delta y

The Gradient and Higher-Order Derivatives

Higher-order partial derivatives

  • Involve repeated differentiation with respect to same or different variables
  • Notation: ∂2f∂x2\frac{\partial^2 f}{\partial x^2}, ∂2f∂x∂y\frac{\partial^2 f}{\partial x \partial y}, fxyf_{xy}, fyxf_{yx} represents higher-order derivatives
  • Mixed partial derivatives result from differentiating with respect to different variables
  • states equality of mixed partials under continuity conditions
  • Applications include analyzing concavity (upward fxx>0f_{xx} > 0, downward fxx<0f_{xx} < 0) and developing Taylor series expansions for multivariable functions

Gradient vector computation and geometry

  • Gradient vector ∇f=(∂f∂x,∂f∂y,∂f∂z)\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right) combines all partial derivatives
  • Properties: perpendicular to level curves/surfaces, points in direction of steepest ascent
  • Relates to directional derivatives through dot product ∇f⋅u\nabla f \cdot \mathbf{u}
  • Facilitates solving optimization problems (finding maxima/minima)
  • Provides normal vectors to surfaces in 3D space
  • Identifies conservative vector fields where curl is zero
  • Components directly correspond to partial derivatives of the function

Key Terms to Review (19)

∂²f/∂x∂y: The term ∂²f/∂x∂y represents the mixed second partial derivative of a function f with respect to the variables x and y. This notation indicates that we first take the partial derivative of the function f with respect to y, and then we take the partial derivative of that result with respect to x. Understanding this concept is crucial because it helps in analyzing how a multivariable function changes as we vary more than one variable, giving insight into the behavior of the function in a multidimensional space.
∂²f/∂x²: The term ∂²f/∂x² represents the second partial derivative of a function f with respect to the variable x. This measures how the rate of change of the function in relation to x itself changes, providing insights into the curvature and behavior of the function at a given point. It is closely related to the concepts of concavity, optimization, and is essential in understanding the geometry of functions of multiple variables.
∂f/∂x: The notation ∂f/∂x represents the partial derivative of a function f with respect to the variable x. It measures how the function f changes as the variable x changes while keeping all other variables constant. This concept is crucial for understanding how multivariable functions behave and is foundational for analyzing gradients and optimization in higher dimensions.
∂f/∂y: The symbol ∂f/∂y represents the partial derivative of a function 'f' with respect to the variable 'y'. This concept measures how the function 'f' changes as 'y' varies, while all other variables remain constant. Partial derivatives are essential for understanding functions of multiple variables, as they help describe the behavior and rates of change in these functions, leading to the formation of gradients that point in the direction of steepest ascent.
∇f: The symbol ∇f represents the gradient of a scalar function f, which is a vector field showing the direction and rate of the steepest ascent of the function. The gradient is a crucial concept that connects partial derivatives with the behavior of functions in multiple dimensions, allowing us to analyze how changes in variables affect the function's output.
Chain Rule: The chain rule is a fundamental concept in calculus that provides a way to compute the derivative of a composite function. It allows you to differentiate functions that are nested within one another by relating the rates of change of the outer function to the rates of change of the inner function. This is especially important when dealing with functions of several variables or vector-valued functions.
Clairaut's Theorem: Clairaut's Theorem states that if the mixed partial derivatives of a function are continuous, then the order of differentiation does not matter. This means that if you take the partial derivative of a function with respect to one variable and then with respect to another, you will get the same result regardless of the order in which you differentiate. This theorem connects the idea of continuity of mixed partials with the equality of those derivatives, which is crucial for understanding how functions behave in multiple dimensions.
Directional Derivative: The directional derivative is a measure of how a function changes as you move in a specific direction from a given point. It generalizes the concept of a derivative to multiple dimensions, allowing you to understand how functions behave when approached from various angles. This concept is deeply linked to partial derivatives and the gradient, which collectively help determine the rate of change of functions in multidimensional spaces.
F_x: The notation f_x represents the partial derivative of a function f with respect to the variable x. This concept focuses on how the function changes as the variable x varies, while keeping all other variables constant. Understanding f_x is crucial for analyzing functions of multiple variables and is foundational for studying gradients and optimization in multivariable calculus.
F_xy: The notation f_xy refers to the mixed partial derivative of a function f with respect to the variables x and y. This derivative measures how the function changes as both x and y vary, holding all other variables constant. Mixed partial derivatives, like f_xy, are essential in understanding the behavior of multivariable functions, especially in optimization and analyzing surface curvature.
F_y: The term f_y represents the partial derivative of a function f with respect to the variable y. It captures how the function changes as y varies while keeping all other variables constant. This concept is essential in understanding how multivariable functions behave and is a foundational component when discussing gradients, as it helps quantify the slope of a function in the direction of one specific variable.
F_yx: The notation f_yx represents the mixed partial derivative of a function f with respect to the variable y first and then x. This means you take the partial derivative of f with respect to y, and then take the partial derivative of that result with respect to x. Understanding this concept is crucial because it highlights how a multivariable function changes in relation to multiple variables, reflecting the interdependency of those variables in determining function behavior.
Gradient vector: The gradient vector is a vector that represents the direction and rate of the steepest ascent of a scalar function. It combines all the partial derivatives of a function into a single vector, which can help in understanding how changes in multiple variables affect the function's output. This concept connects to various aspects, such as how tangent planes approximate surfaces and how directional derivatives provide insight into changing functions along specific paths.
Higher-Order Partial Derivatives: Higher-order partial derivatives are the derivatives of a function taken multiple times with respect to one or more variables. They extend the concept of first-order partial derivatives, which measure the rate of change of a function with respect to a single variable, by allowing for the analysis of how the rate of change itself varies. These derivatives are essential for understanding the behavior of functions in multiple dimensions, especially when studying concavity, curvature, and optimization.
Implicit Differentiation: Implicit differentiation is a technique used to find the derivative of a dependent variable defined implicitly by an equation involving both the dependent and independent variables. Instead of solving for one variable in terms of another, implicit differentiation allows you to differentiate both sides of an equation with respect to the independent variable, applying the chain rule when necessary. This method is especially useful when dealing with equations that cannot be easily solved for one variable.
Lagrange Multipliers: Lagrange multipliers are a strategy used in optimization problems to find the local maxima and minima of a function subject to equality constraints. This method involves introducing an auxiliary variable, known as the Lagrange multiplier, which helps convert a constrained problem into an unconstrained one by incorporating the constraint into the objective function. By utilizing partial derivatives and gradients, this technique allows for the simultaneous solving of equations that define optimal solutions within specified limits.
Mixed partial derivative: A mixed partial derivative is the second derivative of a function with respect to two different variables, taken in succession. This derivative provides insight into how the function changes when varying one variable while holding another variable constant, and then observing how that change affects the first variable. Mixed partial derivatives are crucial for understanding the behavior of functions of multiple variables and are often denoted as \( f_{xy} \) or \( \frac{\partial^2 f}{\partial y \partial x} \).
Partial Derivative: A partial derivative is a derivative of a function of multiple variables with respect to one variable while keeping the other variables constant. This concept is essential in understanding how functions behave in higher dimensions, revealing how changes in one variable affect the function's value without interference from other variables. It plays a critical role in applications like optimization, physics, and economics, as well as in determining the behavior of surfaces and gradients.
Tangent Plane: A tangent plane is a flat surface that touches a curved surface at a single point, representing the local linear approximation of the curved surface at that point. This concept is fundamental in understanding how multivariable functions behave and provides insights into rates of change in multiple dimensions, connecting closely to gradients and surface representations.
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