Partial Derivative Calculator
Pick a two-variable function and a point (x, y) to estimate its partial derivatives ∂f/∂x and ∂f/∂y numerically.
Function f(x,y)
Rate of change of f with respect to x
- 1
f(x+h, y) with h = 1×10⁻⁵
f(1.00001, 2) = 5.00002 - 2
f(x-h, y) with h = 1×10⁻⁵
f(0.99999, 2) = 4.99998 - 3
Numerator = f(x+h,y) − f(x-h,y)
5.00002 − 4.99998 = 0.00004Central difference cancels the first-order error term for better accuracy. - 4
∂f/∂x ≈ numerator ÷ (2h)
0.00004 ÷ 0.00002 = 2
How does this calculator work?
A partial derivative gives the slope of f(x,y) along one axis while the other variable is fixed. This tool evaluates ∂f/∂x and ∂f/∂y at your chosen point using the central-difference formula with step h = 1e-5, yielding accurate numeric slopes for the selected function.
Formula
How this is calculated
A partial derivative measures how a multivariable function changes when only one variable moves and the others are held constant. Choose a function f(x,y) from the list and enter the point coordinates x and y where you want the slopes evaluated.
Each partial derivative is approximated with the central-difference formula. To get ∂f/∂x we hold y fixed and compute [f(x+h, y) − f(x−h, y)] / (2h); for ∂f/∂y we hold x fixed and compute [f(x, y+h) − f(x, y−h)] / (2h). The step size h = 1e-5 is small enough to track the true tangent slope while keeping floating-point round-off low, and the symmetric (central) form cancels the leading error term, giving second-order accuracy.
Inputs are dimensionless real numbers, so the derivatives carry the units of f per unit of x or y. The estimates are extremely accurate for the smooth functions offered here. Edge cases to note: very large x in eˣ·y can overflow, and near sharp features a numeric derivative is only an approximation of the exact analytic value.
Frequently asked questions
It is the derivative of a multivariable function with respect to one variable while treating all other variables as constants, describing the surface slope along that axis.
The central difference [f(x+h) − f(x−h)] / (2h) cancels the first-order error term, so it is second-order accurate and matches the true slope far more closely than a forward or backward difference.
For the smooth functions provided, the numeric estimate typically agrees with the exact analytic derivative to several decimal places, limited only by floating-point round-off.
Also known as
TG we-Calculate Editorial Team. (2026). Partial Derivative Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/partial-derivative-calculator
TG we-Calculate Editorial Team. "Partial Derivative Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/partial-derivative-calculator.
TG we-Calculate Editorial Team, "Partial Derivative Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/partial-derivative-calculator
@misc{wecalculate_partial_derivative_calculator, title = {Partial Derivative Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/partial-derivative-calculator}}, year = {2026}, note = {TG we-Calculate} }
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