Key Concepts of Partial Derivatives

Why This Matters

Partial derivatives are your gateway to understanding how multivariable functions behave—and in Calculus II, you're being tested on your ability to analyze functions that depend on more than one variable simultaneously. This isn't just an extension of single-variable calculus; it's a fundamentally different way of thinking about change. You'll need to master how functions respond when you tweak one variable while holding others fixed, how the gradient captures the direction of steepest increase, and how optimization works when you have multiple inputs to consider.

The concepts here connect directly to applications you'll see on exams: finding tangent planes, optimizing functions with constraints, and understanding how composite functions behave through the multivariable chain rule. Don't just memorize formulas—know why we treat other variables as constants, when Clairaut's theorem applies, and how the gradient relates to directional derivatives. These conceptual connections are exactly what FRQ prompts target.

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Foundations: What Partial Derivatives Measure

Before diving into techniques, you need a rock-solid understanding of what partial derivatives actually represent. A partial derivative isolates the effect of one variable on a function's output while treating all other variables as constants.

Definition of Partial Derivatives

• Measures the rate of change of a function with respect to one variable while all other variables remain fixed—think of slicing a 3D surface with a vertical plane
• Fundamental to multivariable analysis—without this tool, you cannot analyze surfaces, optimize functions, or compute rates of change in higher dimensions
• Conceptually distinct from ordinary derivatives—you're examining change along one axis only, not total change

Notation for Partial Derivatives

• The symbol $\partial$ distinguishes partial derivatives from ordinary derivatives $d$—common notations include $\frac{\partial f}{\partial x}$, $f_x$, and $\partial_x f$
• Higher-order notation uses repeated symbols: $\frac{\partial^2 f}{\partial x^2}$ for second partials, $\frac{\partial^2 f}{\partial x \partial y}$ for mixed partials
• Subscript notation is faster for computation: $f_{xx}$ means differentiate twice with respect to $x$, while $f_{xy}$ means differentiate with respect to $x$ then $y$

Calculating Partial Derivatives

• Treat all other variables as constants—if finding $\frac{\partial}{\partial x}$ of $f(x,y) = x^2y + y^3$, the $y$ terms act like coefficients, giving $2xy$
• Apply standard differentiation rules (product, quotient, chain) exactly as in single-variable calculus—the only difference is what you treat as constant
• Identify your variable clearly before differentiating—careless errors often come from forgetting which variable is "active"

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Higher-Order Behavior: Second Derivatives and Beyond

Higher-order partial derivatives reveal curvature and concavity of multivariable functions—essential for optimization and understanding surface geometry.

Higher-Order Partial Derivatives

• Second partial derivatives like $f_{xx}$ and $f_{yy}$ measure how the rate of change itself changes—analogous to concavity in single-variable calculus
• Used in the second derivative test for classifying critical points as maxima, minima, or saddle points
• Notation matters: $\frac{\partial^2 f}{\partial x^2}$ means differentiate with respect to $x$ twice, giving information about curvature along the $x$-direction

Mixed Partial Derivatives

• Differentiate with respect to different variables in succession—$f_{xy}$ means take $\frac{\partial}{\partial y}$ of $\frac{\partial f}{\partial x}$
• Capture variable interaction—a nonzero mixed partial indicates that changing one variable affects how the function responds to changes in another
• Order of notation: in $\frac{\partial^2 f}{\partial y \partial x}$, you differentiate with respect to $x$ first, then $y$ (read right to left)

Clairaut's Theorem (Equality of Mixed Partials)

• States that $f_{xy} = f_{yx}$ provided both mixed partials are continuous near the point—this is almost always satisfied for functions you'll encounter
• Simplifies calculations dramatically—you can compute mixed partials in whichever order is easier
• When to cite it: if an exam asks you to verify equality of mixed partials, check continuity conditions and invoke Clairaut's theorem

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The Chain Rule and Implicit Differentiation

When functions are composed or defined implicitly, you need specialized techniques. The multivariable chain rule tracks how changes propagate through nested dependencies.

The Chain Rule for Partial Derivatives

• Handles composite functions where intermediate variables depend on other variables—if $z = f(x,y)$ and $x = g(t)$, $y = h(t)$, then $\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$
• Tree diagrams help visualize dependencies—draw arrows from the final output back through intermediate variables to independent variables
• Essential for related rates problems in multiple dimensions and for functions defined parametrically

Implicit Differentiation with Partial Derivatives

• Used when you can't solve explicitly for one variable—given $F(x,y,z) = 0$, find $\frac{\partial z}{\partial x}$ by differentiating both sides
• Formula shortcut: $\frac{\partial z}{\partial x} = -\frac{F_x}{F_z}$ when $F_z \neq 0$—memorize this for speed on exams
• Apply chain rule implicitly—every term with $z$ gets multiplied by $\frac{\partial z}{\partial x}$ when differentiating with respect to $x$

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Gradients and Directional Derivatives

The gradient unifies all first-order partial derivatives into a single vector that points toward steepest ascent. Directional derivatives generalize partial derivatives to arbitrary directions.

Gradient Vector

• Defined as $\nabla f = \langle f_x, f_y \rangle$ (or $\langle f_x, f_y, f_z \rangle$ in 3D)—collects all first partial derivatives into one vector
• Points in the direction of steepest increase—this is the most important geometric fact about gradients
• Magnitude $|\nabla f|$ gives the maximum rate of change of the function at that point

Directional Derivatives

• Measures rate of change in any direction—not just along coordinate axes like partial derivatives
• Computed as $D_{\mathbf{u}}f = \nabla f \cdot \mathbf{u}$ where $\mathbf{u}$ is a unit vector in the desired direction—don't forget to normalize!
• Maximum value equals $|\nabla f|$ (achieved when moving in the gradient direction); minimum is $-|\nabla f|$ (opposite direction)

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Geometric Applications: Tangent Planes and Linearization

Partial derivatives let you construct linear approximations to surfaces—the multivariable analog of tangent lines.

Tangent Planes and Normal Lines

• Tangent plane equation: for $z = f(x,y)$ at point $(a,b,f(a,b))$, the plane is $z - f(a,b) = f_x(a,b)(x-a) + f_y(a,b)(y-b)$
• Normal vector is $\langle f_x, f_y, -1 \rangle$ for explicit surfaces, or $\nabla F$ for implicit surfaces $F(x,y,z) = 0$
• Normal line passes through the point in the direction of the normal vector—used in optimization and geometric analysis

Taylor Series for Multivariable Functions

• Extends polynomial approximation to functions of several variables—the linear terms use first partials, quadratic terms use second partials
• First-order approximation: $f(x,y) \approx f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$—this is the tangent plane equation
• Second-order terms involve $f_{xx}$, $f_{yy}$, and $f_{xy}$—crucial for the second derivative test in optimization

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Optimization and Applications

Partial derivatives are the primary tool for finding extrema of multivariable functions and modeling real-world systems.

Partial Derivatives in Optimization Problems

• Critical points occur where $\nabla f = \mathbf{0}$—set all first partial derivatives equal to zero and solve the system
• Second derivative test uses the discriminant $D = f_{xx}f_{yy} - (f_{xy})^2$: if $D > 0$ and $f_{xx} > 0$, you have a local minimum; $D > 0$ and $f_{xx} < 0$ gives a maximum; $D < 0$ means saddle point
• Boundary analysis required for constrained regions—check critical points and boundary extrema

Applications in Physics and Engineering

• Heat equation and diffusion use $\frac{\partial u}{\partial t}$ and $\nabla^2 u$ (the Laplacian) to model temperature distribution over time
• Fluid dynamics relies on partial derivatives to express conservation laws and flow rates
• Economics uses partial derivatives for marginal analysis—$\frac{\partial C}{\partial x}$ gives marginal cost with respect to one input

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Quick Reference Table

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Self-Check Questions

1. If $f(x,y) = x^3y^2 + \sin(xy)$, what is $f_{xy}$? Verify that $f_{xy} = f_{yx}$ and explain why Clairaut's theorem guarantees this.

2. Compare the gradient vector and the directional derivative: how does knowing $\nabla f$ at a point let you find the rate of change in any direction?

3. For the surface $x^2 + y^2 + z^2 = 9$, use implicit differentiation to find $\frac{\partial z}{\partial x}$. What does this derivative represent geometrically?

4. A critical point has $f_{xx} = 4$, $f_{yy} = 3$, and $f_{xy} = 5$. Calculate the discriminant $D$ and classify the critical point. What would change if $f_{xy} = 2$ instead?

5. Contrast finding the tangent plane to $z = f(x,y)$ at a point versus finding the tangent plane to an implicitly defined surface $F(x,y,z) = 0$. What role does the gradient play in each case?