Partial Derivative Rules

Why This Matters

Partial derivatives are the foundation of Calculus IV. They extend differentiation to functions with multiple inputs. Whenever you analyze a surface, optimize a multivariable function, or study how a physical system responds to changes, you're relying on partial derivatives. The rules here connect directly to gradient vectors, directional derivatives, optimization, and vector field analysis.

You'll be tested on more than computation. Examiners want to see that you understand when to apply each rule and why it works. The chain rule for partials drives related rates problems. Clairaut's theorem saves you time on mixed partials. The gradient ties everything together for optimization. Knowing what concept each rule captures, and when to reach for it, matters just as much as cranking through algebra.

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Foundational Concepts and Notation

Before applying any rules, you need a 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

The formal definition mirrors the single-variable limit definition. For a function $f(x, y)$:

$f_x(a, b) = \lim_{h \to 0} \frac{f(a+h,\, b) - f(a,\, b)}{h}$

• Measures single-variable change: a partial derivative tells you how $f$ changes as one variable varies while all others stay fixed.
• Enables independent analysis of each variable's contribution to the function's behavior.
• Notation $\frac{\partial f}{\partial x}$ uses the curly $\partial$ to distinguish partial differentiation from total (ordinary) differentiation.

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 $\frac{\partial^2 f}{\partial x \partial y}$ mean: differentiate first with respect to $y$, then with respect to $x$. Read right to left in Leibniz notation.
• Higher-order notation extends naturally: $\frac{\partial^3 f}{\partial x^2 \partial y}$ means differentiate once with respect to $y$, then twice with respect to $x$.

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Differentiation Rules for Partial Derivatives

The computational rules from single-variable calculus (product rule, quotient rule, chain rule) all extend to partial derivatives with one key modification: treat every variable except the one you're differentiating with respect to as a constant.

Partial Derivatives of Multivariable Functions

Here's the process:

1. Identify which variable you're differentiating with respect to.
2. Treat all other variables as constants.
3. Apply the usual single-variable differentiation rules (power, product, quotient, exponential, trig, etc.).

Example: For $f(x, y) = x^3 y^2 + \sin(xy)$, finding $f_x$:

• Treat $y$ as a constant.
• $\frac{\partial}{\partial x}(x^3 y^2) = 3x^2 y^2$ (power rule; $y^2$ is just a constant coefficient).
• $\frac{\partial}{\partial x}(\sin(xy)) = y\cos(xy)$ (chain rule; derivative of the inner function $xy$ with respect to $x$ is $y$).
• So $f_x = 3x^2 y^2 + y\cos(xy)$.

Setting all first partials equal to zero simultaneously is how you find critical points of multivariable functions.

Chain Rule for Partial Derivatives

This is where things get genuinely tricky. The chain rule has different forms depending on the dependency structure.

Case 1: Single parameter. If $z = f(x, y)$ where $x = x(t)$ and $y = y(t)$, then:

$\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$

Notice the result is an ordinary derivative $\frac{dz}{dt}$ because $z$ ultimately depends on only one independent variable.

Case 2: Multiple parameters. If $z = f(x, y)$ where $x = x(s, t)$ and $y = y(s, t)$, then:

$\frac{\partial z}{\partial s} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial s} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial s}$

and similarly for $\frac{\partial z}{\partial t}$.

Tree diagrams are genuinely useful here. Draw $z$ at the top, connect it to $x$ and $y$, then connect those to $s$ and $t$. Each path from $z$ down to your target variable contributes one term (multiply along branches, add across paths).

Implicit Differentiation for Multivariable Functions

When a relationship is given as $F(x, y, z) = 0$ and you can't easily solve for $z$, implicit differentiation gives you the partial derivatives directly.

Formula: If $F(x, y, z) = 0$ and $F_z \neq 0$, then:

$\frac{\partial z}{\partial x} = -\frac{F_x}{F_z} \qquad \text{and} \qquad \frac{\partial z}{\partial y} = -\frac{F_y}{F_z}$

Why this works: Differentiate $F(x, y, z) = 0$ with respect to $x$ using the chain rule (treating $z$ as a function of $x$ and $y$), then solve for $\frac{\partial z}{\partial x}$.

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Symmetry and Higher-Order Derivatives

When you take multiple partial derivatives, the order can matter. Clairaut's theorem tells you exactly when you can swap the order, which is most of the time for functions you'll encounter in this course.

Clairaut's Theorem (Equality of Mixed Partials)

Statement: If $f_{xy}$ and $f_{yx}$ are both continuous on an open region, then:

$\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}$

• Computation shortcut: choose whichever differentiation order is easier. The result is identical as long as the continuity condition holds.
• The continuity requirement is almost always satisfied for functions on exams. The exception to watch for is piecewise-defined functions at the boundary point, where the mixed partials can fail to be continuous and may differ.

Higher-Order Partial Derivatives

• Pure second partials like $f_{xx} = \frac{\partial^2 f}{\partial x^2}$ measure concavity in the $x$-direction, just like single-variable second derivatives.
• Mixed partials like $f_{xy}$ capture how the rate of change in one direction varies as you move in another direction. Think of it as measuring "twist."
• The Hessian matrix collects all second partials into a matrix:

$H = \begin{bmatrix} f_{xx} & f_{xy} \\ f_{xy} & f_{yy} \end{bmatrix}$

The second derivative test uses the Hessian's determinant: $D = f_{xx}f_{yy} - (f_{xy})^2$. At a critical point where $\nabla f = \mathbf{0}$:

• $D > 0$ and $f_{xx} > 0$: local minimum
• $D > 0$ and $f_{xx} < 0$: local maximum
• $D < 0$: saddle point
• $D = 0$: test is inconclusive

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Gradient and Directional Analysis

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

Gradient Vector

Definition: For $f(x, y, z)$:

$\nabla f = \left\langle \frac{\partial f}{\partial x},\, \frac{\partial f}{\partial y},\, \frac{\partial f}{\partial z} \right\rangle = \langle f_x,\, f_y,\, f_z \rangle$

• Points toward steepest ascent: moving in the direction of $\nabla f$ increases $f$ as rapidly as possible.
• Magnitude $\|\nabla f\|$ equals the maximum rate of change of $f$ at that point.
• Normal to level curves/surfaces: $\nabla f$ at a point is perpendicular to the level curve (2D) or level surface (3D) passing through that point. This is why the gradient shows up in tangent plane equations.

Directional Derivatives

The directional derivative gives the rate of change of $f$ in any direction, not just along coordinate axes.

$D_{\mathbf{u}}f = \nabla f \cdot \mathbf{u}$

where $\mathbf{u}$ must be a unit vector ($\|\mathbf{u}\| = 1$). If you're given a direction vector that isn't unit length, normalize it first.

Three facts worth memorizing:

• Maximum $D_{\mathbf{u}}f = \|\nabla f\|$, occurring when \mathbf{u}} points in the gradient direction.
• Minimum $D_{\mathbf{u}}f = -\|\nabla f\|$, occurring in the direction opposite the gradient.
• Zero directional derivative occurs when $\mathbf{u}$ is perpendicular to $\nabla f$, meaning you're moving along a level curve/surface.

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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

For a parametric surface $\mathbf{r}(u, v) = \langle x(u,v),\, y(u,v),\, z(u,v) \rangle$, take partials of each component separately:

$\mathbf{r}_u = \left\langle \frac{\partial x}{\partial u},\, \frac{\partial y}{\partial u},\, \frac{\partial z}{\partial u} \right\rangle$

• The vectors $\mathbf{r}_u$ and $\mathbf{r}_v$ are tangent to the surface at each point.
• Their cross product $\mathbf{r}_u \times \mathbf{r}_v$ gives a normal vector to the surface, which you'll need for surface integrals and flux calculations.

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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 $\frac{\partial^2 f}{\partial x \partial y}$ 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?