# Partial Derivatives

Physics I · Multivariable Calculus · https://tryals.app/learn/physics-i/partial-derivatives

A **partial derivative** measures how $f$ changes when one variable moves while the others are held fixed.

$$\frac{\partial f}{\partial x} = \lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h}$$

**Key insight**: Partial derivatives reduce a multivariable problem to a single-variable one by treating all other variables as constants.

**Notation**

| Symbol | Meaning |
| --- | --- |
| $f_x = \frac{\partial f}{\partial x}$ | Partial derivative with respect to $x$ |
| $f_y = \frac{\partial f}{\partial y}$ | Partial derivative with respect to $y$ |
| $f_{xy} = \frac{\partial^2 f}{\partial y \partial x}$ | Second-order mixed partial |

**Geometric interpretation**: $f_x(a,b)$ is the slope of $f$ at $(a,b)$ in the $x$-direction. Likewise, $f_y(a,b)$ is the slope in the $y$-direction.

## Computation & Rules

**Example**: For $f(x,y) = x^{2}y + 3xy^{2}$:

$$f_x = 2xy + 3y^2, \qquad f_y = x^2 + 6xy$$

**Clairaut's theorem**: If $f_{xy}$ and $f_{yx}$ are both continuous, the order of differentiation does not matter:

$$f_{xy} = f_{yx}$$

**Common pitfall**:$\partial f / \partial x$ freezes every other variable, answering a deliberately narrow question. The real-world change of $f$ when several inputs move at once is the **chain rule**'s job, not a single partial's.

## Practice questions

5 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. The temperature in a room is $T(x, y, z, t)$. What experiment does $\partial T/\partial x$ describe?

A. Watch the temperature change over time at a fixed spot
B. Average the temperature along the $x$-axis
C. Measure the total temperature change while walking in any direction
D. Walk along $x$ with a thermometer, everything else frozen

**Answer:** D. Walk along $x$ with a thermometer, everything else frozen

**Why:** A partial derivative is a controlled experiment: vary one input, freeze the rest. Real changes usually vary everything at once, the chain rule will later reassemble the full change from these one-at-a-time answers.

Page: https://tryals.app/practice/physics-i/partial-derivatives/the-temperature-in-a-room-is-t-x-y-z-t-what-experiment-does

### 2. A partial derivative holds all other variables constant. Why is that physically meaningful rather than an artificial restriction?

A. Because the other variables are usually constant in real systems
B. Because experiments are designed the same way: vary one factor
C. Because the total derivative is only an approximation to the partials
D. Because varying two variables at once is mathematically undefined

**Answer:** B. Because experiments are designed the same way: vary one factor

**Why:** Holding variables fixed is exactly experimental control. The total derivative then reassembles the pieces when everything varies together, which is what the chain rule does.

Page: https://tryals.app/practice/physics-i/partial-derivatives/a-partial-derivative-holds-all-other-variables-constant-why-is-that

### 3. Order the steps to find the tangent plane to $z = f(x,y)$ at (a,b).

**Answer:**

1. Compute $f(a,b)$ to find the z-value
2. Compute $f_{x}$(a,b)
3. Compute fᵧ(a,b)
4. Write $z = f(a,b) + f_{x}(a,b)(x-a) + f$ᵧ(a,b)(y−b)
5. Simplify the equation

**Why:** The tangent plane is the best linear approximation to f near (a,b).

Page: https://tryals.app/practice/physics-i/partial-derivatives/order-the-steps-to-find-the-tangent-plane-to-z-f-x-y-at-a-b

### 4. For $f(x,y) = x^{2}y + 3y$, evaluate the partial derivative $f_x$ at the point $(2, 5)$.

**Answer:** 20 (within ±0.01)

**Why:** $f_x = 2xy = 20$. Note how the $3y$ term, constant as far as $x$ is concerned, vanished entirely: partial differentiation sees only the variable it interrogates.

Page: https://tryals.app/practice/physics-i/partial-derivatives/for-f-x-y-x-y-3y-evaluate-the-partial-derivative-fx-at-the

### 5. For well-behaved (smooth) functions, the mixed partials agree: $f_{xy} = f_{yx}$, differentiating in $x$ then $y$ gives the same result as $y$ then $x$.

**Answer:** True

**Why:** True, for continuous second derivatives, the order of partial differentiation is interchangeable (Clairaut/Schwarz theorem). This hidden symmetry is why the Hessian matrix is symmetric and why "exactness" tests work for differential equations.

Page: https://tryals.app/practice/physics-i/partial-derivatives/for-well-behaved-smooth-functions-the-mixed-partials-agree-fxy