Partial Derivative Rules

Why This Matters

Partial derivatives are the foundation of everything you'll do in Calculus IV—they're how we extend the power of differentiation to functions with multiple inputs. When you're analyzing a surface, optimizing a function of several variables, or understanding how physical systems respond to changes, you're using partial derivatives. The rules governing them connect directly to gradient vectors, directional derivatives, optimization, and vector field analysis—all major exam topics.

Here's the key insight: you're being tested on more than just computation. Examiners want to see that you understand when to apply each rule and why it works. The chain rule for partials shows up in related rates problems; Clairaut's theorem saves you time on mixed partials; the gradient ties everything together for optimization. Don't just memorize formulas—know what concept each rule illustrates and when it becomes your go-to tool.

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Foundational Concepts and Notation

Before applying any rules, you need rock-solid understanding of what partial derivatives actually measure and how to communicate them precisely. A partial derivative isolates the rate of change in one direction while treating all other variables as constants.

Definition of Partial Derivatives

• Measures single-variable change—a partial derivative tells you how $f$ changes as one variable varies while all others remain fixed
• Enables independent analysis of each variable's contribution to the function's behavior, essential for understanding multivariable relationships
• Notation $\frac{\partial f}{\partial x}$ explicitly indicates differentiation with respect to $x$, distinguishing it from total derivatives

Partial Derivative Notation

• First-order partials use $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$, or subscript notation like $f_x$, $f_y$
• Mixed second derivatives written as $\frac{\partial^2 f}{\partial x \partial y}$ indicate sequential differentiation with respect to different variables
• Higher-order notation extends naturally: $\frac{\partial^3 f}{\partial x^2 \partial y}$ means differentiate twice with respect to $x$, then once with respect to $y$

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Differentiation Rules for Partial Derivatives

The computational rules you learned in single-variable calculus—product rule, quotient rule, chain rule—all extend to partial derivatives with one crucial modification: treat all variables except the one you're differentiating with respect to as constants.

Partial Derivatives of Multivariable Functions

• Compute each partial separately by treating other variables as constants—for $f(x, y)$, find $\frac{\partial f}{\partial x}$ by treating $y$ as a constant
• Reveals directional behavior showing how the function changes along coordinate axes
• Critical point analysis requires setting all first partials equal to zero simultaneously

Chain Rule for Partial Derivatives

• Composite function differentiation—if $z = f(x, y)$ where $x = x(t)$ and $y = y(t)$, then $\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$
• Tree diagrams help track dependencies when variables depend on multiple intermediate variables
• Essential for implicit relationships and parametric surfaces where variables aren't independent

Implicit Differentiation for Multivariable Functions

• No explicit solving required—differentiate the equation $F(x, y, z) = 0$ directly with respect to your variable of interest
• Formula shortcut: if $F(x, y, z) = 0$, then $\frac{\partial z}{\partial x} = -\frac{F_x}{F_z}$ (provided $F_z \neq 0$)
• Handles constraint equations that define surfaces implicitly, common in optimization problems

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Symmetry and Higher-Order Derivatives

When you take multiple partial derivatives, the order can matter—or not. Clairaut's theorem tells us exactly when we can swap the order, which is most of the time for functions you'll encounter.

Clairaut's Theorem (Equality of Mixed Partials)

• Order doesn't matter when mixed partials are continuous: $\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}$
• Computation shortcut—choose whichever order is easier to calculate; the result is identical
• Continuity requirement is almost always satisfied for functions on exams, but watch for piecewise definitions at boundaries

Higher-Order Partial Derivatives

• Second partials $\frac{\partial^2 f}{\partial x^2}$ measure concavity in the $x$-direction, analogous to single-variable second derivatives
• Mixed partials capture how the rate of change in one direction varies as you move in another direction
• Hessian matrix collects all second partials and determines the nature of critical points (saddle, max, or min)

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Gradient and Directional Analysis

The gradient packages all first partial derivatives into a single vector, unlocking powerful geometric interpretations. The gradient points toward steepest increase and its magnitude tells you how steep.

Gradient Vector

• Definition $\nabla f = \langle f_x, f_y, f_z \rangle$ collects all first-order partial derivatives into one vector
• Points toward steepest ascent—move in the direction of $\nabla f$ to increase $f$ as rapidly as possible
• Magnitude $|\nabla f|$ equals the maximum rate of change of $f$ at that point

Directional Derivatives

• Rate of change in any direction—computed as $D_{\mathbf{u}}f = \nabla f \cdot \mathbf{u}$ where $\mathbf{u}$ is a unit vector
• Maximum value equals $|\nabla f|$ and occurs when $\mathbf{u}$ points in the gradient direction
• Zero directional derivative occurs perpendicular to $\nabla f$, along level curves/surfaces

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Vector-Valued Extensions

When your function outputs a vector instead of a scalar, partial derivatives apply component-by-component. This extends naturally to analyzing vector fields and parametric surfaces.

Partial Derivatives of Vector-Valued Functions

• Component-wise differentiation—for $\mathbf{r}(u, v) = \langle x(u,v), y(u,v), z(u,v) \rangle$, take partials of each component separately
• Tangent vectors $\mathbf{r}_u$ and $\mathbf{r}_v$ span the tangent plane to parametric surfaces
• Applications in physics include velocity fields, force fields, and electromagnetic analysis

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Quick Reference Table

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Self-Check Questions

1. If $f(x, y) = x^2 y + e^{xy}$, which rule do you use to find $\frac{\partial f}{\partial x}$, and what do you treat $y$ as during the calculation?

2. Compare and contrast $\frac{\partial^2 f}{\partial x \partial y}$ and $\frac{\partial^2 f}{\partial y \partial x}$. Under what condition are they equal, and why does this matter computationally?

3. Given $\nabla f = \langle 3, -4 \rangle$ at a point, what is the maximum rate of change of $f$, and in what direction does it occur?

4. When would you choose implicit differentiation over standard partial differentiation? Give an example equation where implicit differentiation is the better approach.

5. FRQ-style: A surface is defined by $z = f(x, y)$. Explain how you would use the gradient to find a vector normal to the level curve $f(x, y) = c$ and a vector normal to the surface itself. What's the relationship between these two normals?