A partial derivative calculator differentiates a function of several variables with respect to one of them at a time. Functions like f(x, y) = x²y + 3xy² describe surfaces rather than curves, and they do not have a single slope — they have a slope in the x-direction, another in the y-direction, and this tool computes each one cleanly.
Quick Answer
To find ∂f/∂x, treat every other variable as a constant and differentiate normally; to find ∂f/∂y, treat x as the constant. For f(x, y) = x²y + 3xy²: ∂f/∂x = 2xy + 3y² and ∂f/∂y = x² + 6xy. At the point (2, 1) these equal 7 and 16 respectively.
What the Curly d Is Telling You
The symbol ∂ is read "partial" and signals exactly the rule above: vary one input, freeze the rest. Geometrically, ∂f/∂x at a point is the slope of the surface when you slice it with a plane of constant y and walk in the x-direction; ∂f/∂y is the slope walking the other way. A hiker on a hillside faces the same situation — the steepness depends on the direction of the step — and partial derivatives measure the two compass directions from which all others are built.
The freezing rule has a consequence students underestimate: a term containing only the frozen variable differentiates to zero (a term like 5y vanishes in ∂f/∂x, just as a constant would), but a term containing both variables keeps the frozen one as a coefficient — 3xy² contributes 3y² to ∂f/∂x, not nothing. Every ordinary rule (power, product, chain, exponential) applies unchanged; the only new skill is deciding, term by term, what counts as a constant this time round. Repeating the process gives second-order partials, and under normal smoothness conditions the mixed partials agree: ∂²f/∂x∂y = ∂²f/∂y∂x, a fact known as Clairaut's theorem and a useful self-check on long calculations.
How to Use the Partial Derivative Calculator
- Enter the multivariable function, using clear brackets so products like xy² cannot be misread.
- Choose the variable to differentiate with respect to — x first, if you need them all.
- Read the partial derivative returned.
- Repeat with each remaining variable to build the full set of first partials.
- To get a numerical rate at a point, substitute the point's coordinates into each partial.
- Cross-check by hand on one term: freeze the other variables and apply the ordinary power rule.
Worked Example: Two Slopes on One Surface
Take f(x, y) = x²y + 3xy². For ∂f/∂x, freeze y: the first term's derivative is 2x·y = 2xy, and the second term is 3y² times x, so its derivative is 3y² — total 2xy + 3y². For ∂f/∂y, freeze x: the first term is x² times y, giving x², and the second is 3x times y², giving 6xy — total x² + 6xy. Now evaluate at (2, 1): ∂f/∂x = 2(2)(1) + 3(1)² = 4 + 3 = 7, and ∂f/∂y = 2² + 6(2)(1) = 4 + 12 = 16. Standing at that point on the surface, it climbs about 7 units per step in the x-direction but 16 per step in the y-direction — the surface is more than twice as steep facing y. The same routine handles other standard forms:
- f = x³ + y³ → ∂f/∂x = 3x², ∂f/∂y = 3y²
- f = e^(xy) → ∂f/∂x = y·e^(xy), ∂f/∂y = x·e^(xy)
- f = sin(x)cos(y) → ∂f/∂x = cos(x)cos(y), ∂f/∂y = −sin(x)sin(y)
- f = x² + y² + z² → partials 2x, 2y and 2z respectively
The same function shows what happens one level deeper, where the mixed-partials check earns its keep. Differentiate ∂f/∂x = 2xy + 3y² once more, this time with respect to y, and you get ∂²f/∂y∂x = 2x + 6y. Now go the other way round: differentiate ∂f/∂y = x² + 6xy with respect to x, giving ∂²f/∂x∂y = 2x + 6y. The two routes agree — at the point (2, 1) both equal 4 + 6 = 10 — exactly as Clairaut's theorem promises for a well-behaved function. The pure second partials, for completeness, are ∂²f/∂x² = 2y and ∂²f/∂y² = 6x, the multivariable analogues of concavity in each direction. When a long exam calculation produces mixed partials that disagree, the theorem tells you there is definitely an error to find, and it is almost always in a coefficient rather than a concept.
These are not classroom decorations. An economist reading a production function Q(K, L) asks for ∂Q/∂L — the extra output from one more unit of labour with capital frozen — and calls it the marginal product of labour; it is a partial derivative wearing a suit. A meteorologist's temperature map T(x, y) has partials that say how fast the air warms as you travel east versus north. In each case the professional question is identical to the textbook one: change this input, hold everything else still, and measure the response. That is all a partial derivative ever claims to do, and doing it reliably is the entire skill.
Where Partial Derivatives Go Wrong
The classic slip is deleting the frozen variable entirely instead of carrying it as a coefficient — treating 3xy² as if it were just 3x when differentiating by x. The mirror-image slip is differentiating the frozen variable too, which produces a full derivative of a multivariable function, a thing that does not exist in this simple form. Third, with functions like e^(xy) or sin(xy), forgetting the chain rule factor: the partial of e^(xy) with respect to x is y·e^(xy), and dropping the y is the same century-old error as in single-variable work. Fourth, when evaluating at a point, substitute into both variables of the partial you computed — leaving a stray y in a "numerical" answer means a substitution was missed. Finally, sign discipline with negative terms and trig derivatives matters more here simply because there are twice as many lines to go wrong in.
From Two Slopes to Any Direction
First partials are the raw material for the gradient vector and everything built on it. The Directional Derivative Calculator takes exactly these two numbers and answers the natural next question — how steep is the surface in a direction that is neither pure x nor pure y? — while the single-variable Derivative Calculator covers the one-input case this topic generalises. Explore the rest of the multivariable set under Math Calculators.