5️⃣Multivariable Calculus Unit 3 – Partial Derivatives

Key Concepts

• Partial derivatives measure the rate of change of a function with respect to one variable while holding other variables constant
• Multivariable functions map multiple input variables to a single output value (e.g., $f(x, y) = x^2 + y^2$)
• The notation $\frac{\partial f}{\partial x}$ represents the partial derivative of $f$ with respect to $x$
• Higher-order partial derivatives involve taking partial derivatives of partial derivatives
• The chain rule for partial derivatives allows for finding the derivative of a composite function
  • Requires applying the chain rule to each variable separately
• Partial derivatives have applications in various fields, including physics (e.g., heat transfer) and economics (e.g., marginal cost)
• Understanding the geometric interpretation of partial derivatives helps visualize their meaning

Functions of Multiple Variables

• A function of multiple variables, such as $f(x, y)$, assigns a unique output value to each pair of input values $(x, y)$
• The domain of a multivariable function is a subset of $\mathbb{R}^n$, where $n$ is the number of input variables
• Multivariable functions can be represented graphically as surfaces in three-dimensional space (for functions of two variables)
• Level curves (or contour plots) are two-dimensional representations of multivariable functions
  • They show the sets of points where the function has a constant value
• Continuity and differentiability of multivariable functions can be defined using limits and partial derivatives
• Examples of multivariable functions include:
  • $f(x, y) = x^2 - y^2$ (hyperbolic paraboloid)
  • $g(x, y, z) = x + y + z$ (plane in three-dimensional space)

Partial Derivatives: Definition and Notation

• The partial derivative of a function $f(x, y)$ with respect to $x$ is denoted as $\frac{\partial f}{\partial x}$ or $f_x(x, y)$
• To find $\frac{\partial f}{\partial x}$, treat all other variables (e.g., $y$) as constants and differentiate $f$ with respect to $x$
• The partial derivative of $f(x, y)$ with respect to $y$ is denoted as $\frac{\partial f}{\partial y}$ or $f_y(x, y)$
• Partial derivatives can be evaluated at specific points by substituting the values of the variables
• For example, given $f(x, y) = x^2y + xy^2$:
  • $\frac{\partial f}{\partial x} = 2xy + y^2$
  • $\frac{\partial f}{\partial y} = x^2 + 2xy$
• Partial derivatives can be used to find the instantaneous rate of change of a function in a specific direction

Geometric Interpretation

• Partial derivatives can be interpreted geometrically as the slope of a tangent line to a surface in a specific direction
• For a function $f(x, y)$, $\frac{\partial f}{\partial x}$ represents the slope of the tangent line in the $x$-direction (parallel to the $x$-axis)
• Similarly, $\frac{\partial f}{\partial y}$ represents the slope of the tangent line in the $y$-direction (parallel to the $y$-axis)
• The direction of the tangent line is determined by holding one variable constant and allowing the other to vary
• The magnitude of the partial derivative indicates the steepness of the surface in the corresponding direction
• At a given point, the gradient vector $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)$ points in the direction of the steepest ascent
• The gradient vector is perpendicular to the level curve at the point of interest

Higher-Order Partial Derivatives

• Higher-order partial derivatives are obtained by taking partial derivatives of partial derivatives
• The notation $\frac{\partial^2 f}{\partial x^2}$ or $f_{xx}$ represents the second partial derivative of $f$ with respect to $x$
• Mixed partial derivatives involve taking partial derivatives with respect to different variables in succession
  • For example, $\frac{\partial^2 f}{\partial x \partial y}$ or $f_{xy}$ represents taking the partial derivative of $f$ with respect to $x$, then taking the partial derivative of the result with respect to $y$
• Clairaut's theorem states that mixed partial derivatives are equal if the function is continuous and the partial derivatives are continuous
  • In other words, $f_{xy} = f_{yx}$ under these conditions
• Higher-order partial derivatives can provide information about the curvature and concavity of a surface

The Chain Rule for Partial Derivatives

• The chain rule for partial derivatives allows for finding the derivative of a composite function
• If $f(x, y)$ is a composite function, such as $f(x, y) = g(u(x, y), v(x, y))$, then:
  • $\frac{\partial f}{\partial x} = \frac{\partial g}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial g}{\partial v} \frac{\partial v}{\partial x}$
  • $\frac{\partial f}{\partial y} = \frac{\partial g}{\partial u} \frac{\partial u}{\partial y} + \frac{\partial g}{\partial v} \frac{\partial v}{\partial y}$
• The chain rule requires applying the product rule and the chain rule to each variable separately
• It is essential to identify the inner and outer functions and their respective variables
• For example, if $f(x, y) = \sin(x^2 + y^2)$, then:
  • $\frac{\partial f}{\partial x} = \cos(x^2 + y^2) \cdot 2x$
  • $\frac{\partial f}{\partial y} = \cos(x^2 + y^2) \cdot 2y$

Applications in Physics and Economics

• Partial derivatives have numerous applications in various fields, including physics and economics
• In physics, partial derivatives are used to describe phenomena such as heat transfer, fluid dynamics, and electromagnetic fields
  • For example, the heat equation $\frac{\partial T}{\partial t} = \alpha \nabla^2 T$ involves partial derivatives of temperature with respect to time and space
• In economics, partial derivatives are used to analyze marginal quantities, such as marginal cost and marginal revenue
  • Marginal cost is the partial derivative of the total cost function with respect to the quantity produced
  • Marginal revenue is the partial derivative of the total revenue function with respect to the quantity sold
• Partial derivatives can also be used to optimize multivariable functions, such as finding the minimum cost or maximum profit
• Lagrange multipliers, a technique involving partial derivatives, is used to optimize functions subject to constraints

Common Pitfalls and Tips

• When taking partial derivatives, it is crucial to treat all variables except the one being differentiated as constants
• Be careful with the chain rule for partial derivatives, as it requires applying the chain rule to each variable separately
• Remember that the order of mixed partial derivatives matters unless the function is continuous and the partial derivatives are continuous (Clairaut's theorem)
• When evaluating partial derivatives at specific points, substitute the values only after taking the partial derivatives
• Sketch level curves or contour plots to visualize the behavior of a multivariable function
• Use the geometric interpretation of partial derivatives to understand their meaning and direction
• Practice applying the chain rule for partial derivatives with various composite functions to develop proficiency
• Pay attention to the units of partial derivatives, especially in applications like physics and economics