∞Calculus IV Unit 3 – Partial Derivatives

What's the Big Idea?

• Partial derivatives extend the concept of single-variable derivatives to functions of multiple variables
• Allow us to analyze how a function changes with respect to one variable while holding other variables constant
• Enable us to study the behavior of functions in higher dimensions (3D and beyond)
• Fundamental tool for optimization problems involving multiple variables
• Used extensively in fields such as physics, engineering, and economics to model complex systems
  • Help describe the behavior of fluids, heat transfer, and electromagnetic fields
  • Crucial for analyzing the sensitivity of financial models to various parameters
• Provide a way to visualize and interpret the gradient and directional derivatives of a function

Key Concepts

• Partial derivative: the derivative of a function with respect to one of its variables, treating other variables as constants
• Second-order partial derivatives: partial derivatives of partial derivatives, denoted by $\frac{\partial^2 f}{\partial x^2}$, $\frac{\partial^2 f}{\partial x \partial y}$, etc.
• Mixed partial derivatives: second-order partial derivatives obtained by differentiating with respect to different variables in succession
  • Clairaut's theorem states that mixed partial derivatives are equal if they are continuous
• Gradient: a vector that points in the direction of the greatest rate of increase of a function, denoted by $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$
• Directional derivative: the rate of change of a function in a specific direction, given by the dot product of the gradient and a unit vector in that direction
• Tangent plane: a plane that best approximates a surface at a given point, determined by the gradient at that point
• Critical points: points where all partial derivatives are zero, which can be local minima, maxima, or saddle points

Notation and Definitions

• Partial derivative notation: $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$, etc., where the symbol $\partial$ is used instead of $d$ to emphasize that only one variable is changing
• Higher-order partial derivative notation: $\frac{\partial^2 f}{\partial x^2}$, $\frac{\partial^2 f}{\partial x \partial y}$, etc., where the order of differentiation is indicated by the superscript and the variables
• Gradient notation: $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$, where $\nabla$ is the del operator
• Directional derivative notation: D_\vec{u} f = \nabla f \cdot \vec{u}, where $\vec{u}$ is a unit vector in the desired direction
• Hessian matrix: a square matrix of second-order partial derivatives, used to analyze the local behavior of a function and determine the nature of critical points
  • Denoted by $H(f) = \begin{bmatrix} \frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x \partial y} \\ \frac{\partial^2 f}{\partial y \partial x} & \frac{\partial^2 f}{\partial y^2} \end{bmatrix}$ for a function of two variables

Techniques and Methods

• Computing partial derivatives using the limit definition: $\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h}$
• Applying rules of differentiation (sum, product, quotient, chain rules) to find partial derivatives
  • Example: if $f(x, y) = x^2 y + \sin(xy)$, then $\frac{\partial f}{\partial x} = 2xy + y\cos(xy)$ and $\frac{\partial f}{\partial y} = x^2 + x\cos(xy)$
• Using the gradient to find the direction of steepest ascent or descent
  • The gradient always points in the direction of the greatest rate of increase
  • To find the direction of steepest descent, take the negative of the gradient
• Determining the tangent plane to a surface at a point using the gradient
  • The equation of the tangent plane at $(x_0, y_0, f(x_0, y_0))$ is $z - f(x_0, y_0) = \frac{\partial f}{\partial x}(x_0, y_0)(x - x_0) + \frac{\partial f}{\partial y}(x_0, y_0)(y - y_0)$
• Finding critical points by setting all partial derivatives equal to zero and solving the resulting system of equations
• Classifying critical points using the second derivative test or the eigenvalues of the Hessian matrix
  • If the Hessian is positive definite (all eigenvalues > 0), the point is a local minimum
  • If the Hessian is negative definite (all eigenvalues < 0), the point is a local maximum
  • If the Hessian has both positive and negative eigenvalues, the point is a saddle point

Applications in Real Life

• Optimization problems in economics, such as finding the optimal production levels to maximize profit or minimize cost
  • Example: a company producing two products with limited resources can use partial derivatives to determine the optimal mix of products
• Modeling heat transfer and fluid dynamics in engineering and physics
  • Partial derivatives help describe the flow of heat or fluids in various directions
  • Used in the design of heat exchangers, airfoils, and other systems involving heat or fluid flow
• Analyzing the sensitivity of financial models to changes in various parameters (interest rates, volatility, etc.)
  • Partial derivatives help quantify the impact of small changes in inputs on the output of a financial model
• Studying the behavior of electric and magnetic fields in electromagnetism
  • Maxwell's equations, which govern electromagnetic phenomena, are expressed using partial derivatives
• Investigating the properties of materials under stress and strain in continuum mechanics
  • Partial derivatives are used to formulate the equations describing the deformation and stress distribution in materials
• Optimizing the design of structures, such as bridges and buildings, to minimize weight or maximize strength
  • Partial derivatives help find the optimal shape and dimensions of structural components

Common Pitfalls

• Forgetting to treat other variables as constants when computing partial derivatives
  • When finding $\frac{\partial f}{\partial x}$, treat y and z as constants, and vice versa
• Misapplying the chain rule when dealing with composite functions
  • If $f(x, y) = g(x^2 + y^2)$, then $\frac{\partial f}{\partial x} = g'(x^2 + y^2) \cdot 2x$, not just $g'(x^2 + y^2)$
• Confusing the order of differentiation for mixed partial derivatives
  • In general, $\frac{\partial^2 f}{\partial x \partial y} \neq \frac{\partial^2 f}{\partial y \partial x}$ unless the mixed partial derivatives are continuous
• Misinterpreting the meaning of the gradient
  • The gradient points in the direction of steepest ascent, not necessarily the direction of the function's maximum value
• Incorrectly classifying critical points without considering the second derivative test or the eigenvalues of the Hessian matrix
  • A critical point where all partial derivatives are zero is not necessarily a local minimum or maximum
• Overlooking the importance of the domain when analyzing functions of multiple variables
  • The behavior of a function and the existence of partial derivatives can change depending on the domain of the function

Practice Problems

1. Find the partial derivatives of $f(x, y) = x^3 y^2 + e^{xy} - \ln(x^2 + y^2)$
2. Determine the gradient of $g(x, y, z) = x^2 yz - \sin(xyz)$ at the point $(1, \pi/2, -1)$
3. Find the directional derivative of $h(x, y) = x^2 - 2xy + 3y^2$ at $(1, 1)$ in the direction of the vector $\vec{u} = \frac{1}{\sqrt{2}}(1, 1)$
4. Find the equation of the tangent plane to the surface $z = x^2 + y^2 - 2xy$ at the point $(1, -1, 2)$
5. Classify the critical point $(0, 0)$ of the function $f(x, y) = x^3 - 3xy^2$
6. Show that the mixed partial derivatives of $f(x, y) = \frac{xy}{x^2 + y^2}$ are equal at the point $(1, 1)$
7. Find the maximum value of the function $f(x, y) = 4x - x^2 - y^2$ on the domain $\{(x, y) | x \geq 0, y \geq 0, x + y \leq 2\}$
8. A manufacturer produces two types of products, A and B. The profit per unit for product A is \20$, and for product B, it is$$30$. The production of each unit of A requires 2 hours of machine time and 3 hours of labor, while each unit of B requires 3 hours of machine time and 2 hours of labor. The manufacturer has a total of 1200 hours of machine time and 1500 hours of labor available. How many units of each product should be produced to maximize profit?

Connections to Other Topics

• Partial derivatives are a fundamental concept in multivariable calculus, which builds upon single-variable calculus
  • They extend the ideas of limits, continuity, and differentiability to functions of multiple variables
• The gradient and directional derivatives are closely related to the concept of the total derivative in single-variable calculus
  • The total derivative measures the rate of change of a function in the direction of a specific vector, while the gradient provides the direction of the greatest rate of change
• Partial derivatives are used in the formulation and solution of partial differential equations (PDEs)
  • PDEs describe phenomena that depend on multiple variables, such as heat transfer, wave propagation, and fluid dynamics
• The Hessian matrix and the second derivative test for functions of multiple variables are analogous to the second derivative test for single-variable functions
  • They help determine the nature of critical points (minima, maxima, or saddle points)
• Partial derivatives play a crucial role in vector calculus, which deals with vector fields and their properties
  • The divergence and curl of a vector field, which measure the field's "spreading" and "rotation," respectively, are defined using partial derivatives
• The concept of partial derivatives is essential for understanding and applying the methods of optimization in multiple dimensions
  • Techniques such as the method of Lagrange multipliers and the Karush-Kuhn-Tucker conditions rely on partial derivatives to find optimal solutions subject to constraints