What a Partial Derivative Is
A partial derivative measures how a function changes when you move one variable and freeze all the others. For f(x, y), the symbol ∂f/∂x means: treat y as a constant number and differentiate in x as usual. The curly ∂ (Jacobi’s notation) warns that other variables exist.
Picture a hillside. ∂f/∂x is the slope you feel walking due east, ∂f/∂y the slope walking due north. The gradient ∇f = (∂f/∂x, ∂f/∂y) points the way that is steepest uphill.
f(x, y) = x²y + 3xy²
- For ∂f/∂x treat y as constant: d/dx[x²y] = 2xy and d/dx[3xy²] = 3y².
- For ∂f/∂y treat x as constant: d/dy[x²y] = x² and d/dy[3xy²] = 6xy.
- At (1, 2): fx = 2(1)(2) + 3(4) = 16 and fy = 1 + 6(2) = 13.
Answer: fx = 2xy + 3y², fy = x² + 6xy
Second Partials and Clairaut’s Theorem
Differentiating twice gives four second partials: fxx, fyy and the mixed pair fxy, fyx. Clairaut’s theorem says the mixed pair are equal whenever they are continuous, so the order does not matter.
f = x²y + 3xy², find fxy and fyx
- fx = 2xy + 3y², so fxy = 2x + 6y.
- fy = x² + 6xy, so fyx = 2x + 6y.
Answer: Both equal 2x + 6y
This is a built-in check. If you ever get two different mixed partials, you made an arithmetic slip.
Gradient, Directional Derivative and Tangent Plane
| Quantity | Formula | Meaning |
|---|---|---|
| Gradient | ∇f = (fx, fy) | Direction of fastest increase |
| Directional derivative | Duf = ∇f · u (u a unit vector) | Slope in any chosen direction |
| Tangent plane | z = f(a,b) + fx(x−a) + fy(y−b) | Best flat approximation near (a, b) |
| Hessian | [[fxx, fxy],[fxy, fyy]] | Curvature, used to classify critical points |
f = x²y + 3xy² at (1, 2), direction toward (3, 4)
- ∇f(1, 2) = (16, 13).
- Direction vector (3, 4) has length 5, so u = (3/5, 4/5).
- Duf = 16·(3/5) + 13·(4/5) = (48 + 52) ÷ 5.
Answer: 20
Critical Points and the Second-Derivative Test
Set both first partials to zero and solve. Then compute D = fxxfyy − (fxy)². If D > 0 and fxx > 0 it is a local minimum, D > 0 and fxx < 0 a local maximum, D < 0 a saddle, and D = 0 settles nothing.
f = x² + y² − 2x − 4y
- fx = 2x − 2 = 0 gives x = 1; fy = 2y − 4 = 0 gives y = 2.
- fxx = 2, fyy = 2, fxy = 0, so D = 4 > 0 and fxx > 0.
- f(1, 2) = 1 + 4 − 2 − 8 = −5.
Answer: Minimum of −5 at (1, 2)
We ran the calculator on f = xy·e−x²−y². It reports a saddle at the origin and two maxima and two minima at (±0.7071, ±0.7071) with values ±0.18394, matching the hand calculation e−1/2.
Where Partial Derivatives Show Up
- Thermodynamics: for an ideal gas P = nRT/V, (∂P/∂T) at constant volume is nR/V.
- Economics: marginal cost of one input while holding the other fixed.
- Machine learning: gradient descent updates each weight with its partial derivative of the loss.
- Physics: the heat and wave equations are written entirely in partials.
What We Think Is Overrated, Underrated and Wrong
A Worked Example With a Mixed Partial
Let f(x, y) = x²y + sin(xy). Then ∂f/∂x = 2xy + y cos(xy) and ∂f/∂y = x² + x cos(xy). Differentiating ∂f/∂x with respect to y gives the mixed partial 2x + cos(xy) − xy sin(xy), and differentiating ∂f/∂y with respect to x gives the same expression. That equality is Clairaut’s theorem, which holds when the second partials are continuous.
For f = x² + y² at (1, 2) the gradient is (2x, 2y) = (2, 4), with magnitude √20 = 4.472, the steepest rate of increase at that point.
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: memorising partial-derivative notation. ∂f/∂x just means “differentiate with y held constant”.
Underrated: the gradient. One vector holds every first partial and points uphill.
Plain wrong: treating y as a variable when differentiating with respect to x. It behaves like a number.
Partial Derivative Mistakes That Cost Marks
- Differentiating y as if it depended on x.
- Forgetting the chain rule inside, e.g. ∂/∂x exy = y·exy.
- Ignoring D = 0, which is inconclusive.
- Using degrees in trig functions.
- Stopping at critical points without classifying them.
Why Students Get Stuck on Partial Derivatives, and What a Good Calculator Should Do
Searching for help with partial derivatives usually starts late at night with one problem that will not work out. There are several variables, and it is easy to differentiate the wrong one. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to hold the other variables constant and differentiate, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Differentiating with respect to the wrong variablePartial derivative of x^2 * y with respect to x↗
- Treating the other variable as changingPartial derivative of x^2 * y with respect to y↗
- Mixing up the order of mixed partialsPartial derivative of sin(x*y) with respect to x↗
- Missing the chain rule inside a functionPartial derivative of x^3 + 3*x*y^2 with respect to y↗
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants each variable treated clearly
- Wants second and mixed partials
- Wants a gradient at a point
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Partial derivative of x^2 * y with respect to x↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.