The formula this page uses
∂f/∂x = lim[h→0] ( f(x + h, y) − f(x, y) ) / h, with y held constant
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Partial Derivative Calculator
Solve partial derivative problems with clear steps, notation, and a final check.
d/dx
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Calculus
Partial Derivative Calculator explained
The short version
- A partial derivative differentiates with respect to one variable while every other variable is frozen as a constant.
- The curly ∂ symbol is the signal that other variables exist and are being held still.
- Mixed second partials are equal for the functions you meet in class, so f_xy and f_yx are a free check on your work.
What each part means
SymbolWhat it means
f(x, y) — The functionA rule with two or more inputs, such as a surface height or a temperature.
∂f/∂x — Partial in xSlope of the surface in the x direction only, measured as a rise per unit of x.
∂f/∂y — Partial in ySlope in the y direction only. Usually a completely different number.
f_xy — Mixed second partialDifferentiate by x, then by y. Written ∂²f/∂y∂x.
Show your work: a full example
- The functionf(x, y) = x³y² + 5x − 4y
- Differentiate by x, treating y² as a fixed number∂f/∂x = 3x²y² + 5
- Note that −4y vanished−4y has no x in it, so it behaves like a constant and differentiates to 0
- Differentiate by y, treating x³ as a fixed number∂f/∂y = 2x³y − 4
- Evaluate ∂f/∂x at (2, 3)3(2)²(3)² + 5 = 3(4)(9) + 5 = 108 + 5 = 113
- Evaluate ∂f/∂y at (2, 3)2(2)³(3) − 4 = 2(8)(3) − 4 = 48 − 4 = 44
- Read itat (2, 3) the surface climbs 113 per unit east and 44 per unit north
A second, different case
- A different case: second-order partials of the same functionf(x, y) = x³y² + 5x − 4y
- Differentiate ∂f/∂x = 3x²y² + 5 again by xf_xx = 6xy²
- Differentiate ∂f/∂y = 2x³y − 4 again by yf_yy = 2x³
- Now differentiate ∂f/∂x by y insteadf_xy = ∂/∂y (3x²y² + 5) = 6x²y
- And differentiate ∂f/∂y by xf_yx = ∂/∂x (2x³y − 4) = 6x²y
- They match, as Clairaut's theorem promisesf_xy = f_yx = 6x²y
- Evaluate all three at (2, 3)f_xx = 6(2)(9) = 108, f_yy = 2(8) = 16, f_xy = 6(4)(3) = 72
Copy-ready example
f(x, y) = x³y² + 5x − 4y
The function
Questions about the Partial Derivative Calculator
What does ∂f/∂x mean on a picture?
Slice the surface with the plane y = 3, which leaves an ordinary curve, and measure that curve's slope. ∂f/∂y slices the other way with x = 2. The two slices through the same point usually have very different steepness, here 113 against 44.
Are f_xy and f_yx always equal?
For every function in a normal course, yes, and Clairaut's theorem says so whenever the second partials are continuous. Counterexamples exist but are deliberately pathological, so a mismatch in your homework means an arithmetic slip.
How do the partials combine into the gradient?
Stack them: ∇f = (∂f/∂x, ∂f/∂y). At (2, 3) that is (113, 44), a vector pointing in the steepest uphill direction, with length √(113² + 44²) giving the steepest slope.
Do I still need the product and chain rules?
Yes, exactly as in one variable. For sin(xy) differentiated by x, the inside function xy has x-derivative y, so the chain rule gives y·cos(xy), which is the row in the table above.