Partial Derivative Calculator
Compute the partial derivative of a 2D function with respect to x or y.
Choose a function form, enter coefficients, and evaluate at any point.
A partial derivative is the rate of change of a multivariable function with respect to one variable, with all other variables held constant. Written as ∂f/∂x or ∂f/∂y, it answers the question: “If I bump x by a tiny amount and freeze y, how fast does f respond?”
The mechanics are exactly the same as ordinary differentiation. Treat the other variable as a constant (number) and apply the standard rules.
This calculator supports three function forms:
Separable sum: f(x,y) = ax^p + by^q. ∂f/∂x = pax^(p-1) (the by^q term has no x, so it differentiates to 0). ∂f/∂y = qby^(q-1).
Product: f(x,y) = ax^p * y^q. ∂f/∂x = pax^(p-1) * y^q (y^q is treated as a constant multiplier). ∂f/∂y = qax^p * y^(q-1).
Trigonometric: f(x,y) = asin(bx) + ccos(dy). ∂f/∂x = abcos(bx). ∂f/∂y = -cdsin(dy).
Worked example: f(x,y) = 3x^2 * y^4 at point (1, 2). Differentiating with respect to x: ∂f/∂x = 6x * y^4 = 6(1)(16) = 96. With respect to y: ∂f/∂y = 12x^2 * y^3 = 12(1)(8) = 96. Both partials equal 96 at this particular point, which is coincidence not pattern.
Partial derivatives are the building blocks of multivariable calculus — gradients, divergence, curl, the chain rule for several variables, and the Jacobian matrix all start here. They appear constantly in thermodynamics (where you constantly hold one variable fixed while varying another), economics (marginal cost with output held constant), and machine learning (every weight update is a partial derivative of the loss).
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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