Chapter 4.3: Partial Derivatives

This guide covers the fundamental concepts of partial differentiation for functions of several variables, including definitions, notations, and higher-order derivatives.

Learning Objectives

By the end of this study guide, you should be able to:

• Calculate the partial derivatives of a function of two variables using both the limit definition and differentiation rules.
• Extend these calculations to functions of more than two variables.
• Determine higher-order partial derivatives, including mixed partials.
• Explain the concept of a partial differential equation (PDE) and identify examples.

Key Terms & Glossary

The "Big Idea"

[!IMPORTANT] In single-variable calculus, the derivative $f'(x)$ represents the slope of a tangent line. In multivariable calculus, the surface $z = f(x, y)$ has infinitely many tangent lines at a point. Partial derivatives simplify this by looking at the slope in directions parallel to the coordinate axes ($x$ and $y$). It's the equivalent of slicing a 3D surface with a plane to create a 2D curve, then finding the slope of that curve.

Formula / Concept Box

Limit Definitions of Partial Derivatives

Hierarchical Outline

1. Introduction to Partial Differentiation
   • Notation Shift: Moving from $d/dx$ to $\partial/\partial x$ (the "partial" symbol).
   • Geometric Interpretation: Slope of the trace of the surface on a plane constant to an axis.
2. Calculation Techniques
   • Rule of Thumb: Treat all variables except the one being differentiated as constants.
   • Applying Power/Product/Chain Rules: These rules from Calc I still apply exactly the same way.
3. Functions of Three or More Variables
   • For $w = f(x, y, z)$, find $f_z$ by holding $x$ and $y$ constant.
4. Higher-Order Partial Derivatives
   • Second Order: $f_{xx}$ (differentiate twice by $x$), $f_{yy}$ (twice by $y$).
   • Mixed Partials: $f_{xy}$ (differentiate by $x$ then by $y$) and $f_{yx}$.

Visual Anchors

Decision Logic for Partial Derivatives

Geometric Interpretation (Intersection of Planes)

Definition-Example Pairs

• Term: Partial Derivative Calculation
• Definition: The process of applying standard derivative rules to one variable while treating others as numbers.
• Real-world Example: If $f(x, y) = 3x^2y + 5y^3$, then $f_x$ treats $y$ as a constant (like the number 5).
  • $f_x = \frac{\partial}{\partial x}(3x^2y) + \frac{\partial}{\partial x}(5y^3) = 6xy + 0 = 6xy$.

Worked Examples

Example 1: Two Variables

Find $f_x$ and $f_y$ for $f(x, y) = x^2 \sin(y)$.

1. To find $f_x$: Treat $\sin(y)$ as a constant. $f_x = \frac{\partial}{\partial x}[x^2] \cdot \sin(y) = 2x \sin(y)$
2. To find $f_y$: Treat $x^2$ as a constant. $f_y = x^2 \cdot \frac{\partial}{\partial y}[\sin(y)] = x^2 \cos(y)$

Example 2: Higher-Order Mixed Partials

Find $f_{xy}$ for $f(x, y) = e^{xy}$.

1. First, find $f_x$: Using the chain rule: $f_x = e^{xy} \cdot \frac{\partial}{\partial x}(xy) = y e^{xy}$.
2. Next, find the derivative of $f_x$ with respect to $y$: Use the product rule on $y \cdot e^{xy}$: $f_{xy} = \frac{\partial}{\partial y}[y] \cdot e^{xy} + y \cdot \frac{\partial}{\partial y}[e^{xy}]$ $f_{xy} = (1)e^{xy} + y(xe^{xy}) = e^{xy}(1 + xy)$

Checkpoint Questions

1. Calculate $f_y$ for $f(x, y) = 4x^3 + 2xy^2 - y^4$.
   • (Answer: $4xy - 4y^3$)
2. If $w = xyz$, what is $w_z$?
   • (Answer: $xy$)
3. True/False: To calculate a partial derivative with respect to $x$, we must use the limit definition every time.
   • (Answer: False, we can use standard differentiation rules.)
4. What is the second-order partial derivative $f_{xx}$ if $f(x, y) = x^5y$?
   • (Answer: $20x^3y$)

[!TIP] When calculating mixed partials like $f_{xy}$, remember the order of operations: $f_{xy}$ usually means differentiate with respect to $x$ first, then $y$ (though Clairaut's Theorem states $f_{xy} = f_{yx}$ for most smooth functions you will encounter).