Partial Derivative Calculator
Computes the partial derivative of a two-variable power function f(x,y) = aₓ·xᵐ·yⁿ with respect to either x or y at a specific point. Useful for multivariable calculus, gradient calculations, and optimization problems.
About this calculator
A partial derivative measures how a multivariable function changes when only one variable varies while the others are held constant. For a function f(x, y) = aₓ·xᵐ·yⁿ, the partial derivative with respect to x is ∂f/∂x = aₓ·m·x^(m−1)·yⁿ, treating y as a constant. Similarly, ∂f/∂y = aᵧ·n·xᵐ·y^(n−1), treating x as a constant. This follows directly from the standard power rule of single-variable differentiation applied to one variable at a time. Partial derivatives are the building blocks of the gradient vector ∇f = (∂f/∂x, ∂f/∂y), which points in the direction of steepest ascent. They appear throughout physics, economics, and machine learning wherever quantities depend on multiple inputs.
How to use
Consider f(x, y) = 3x²y³. Set coefficientX = 3, xExponent = 2, yExponent = 3, and choose variable = x. Evaluate at x = 2, y = 1. The formula gives ∂f/∂x = 3 · 2 · x^(2−1) · y³ = 6x·y³. At (2, 1): 6 · 2 · 1³ = 12. Now try variable = y with coefficientY = 3: ∂f/∂y = 3 · 3 · x² · y^(3−1) = 9x²y². At (2, 1): 9 · 4 · 1 = 36. So the function rises 12 units per unit increase in x and 36 units per unit increase in y at that point.
Frequently asked questions
What is a partial derivative and when do I need to calculate one?
A partial derivative tells you the instantaneous rate of change of a multivariable function with respect to one variable while all others remain fixed. You need it whenever a quantity depends on more than one input — for example, how profit changes with price while holding production constant, or how temperature changes in the x-direction in a 2D heat map. In machine learning, partial derivatives of the loss function with respect to each model parameter are computed during backpropagation. Partial derivatives are also used to find critical points (maxima, minima, saddle points) of multivariable functions.
How is a partial derivative different from a total derivative?
A partial derivative isolates the effect of one variable by holding all others constant, giving ∂f/∂x. A total derivative accounts for the fact that other variables may themselves depend on x, combining all indirect effects via the chain rule: df/dx = ∂f/∂x + (∂f/∂y)(dy/dx). For truly independent variables the two coincide, but when variables are linked — as on a constrained curve — they differ. In most multivariable calculus courses, partial derivatives are introduced first because they are conceptually simpler, with total derivatives covered later alongside implicit differentiation and the chain rule.
Why does the power rule apply when taking a partial derivative?
When differentiating f(x, y) = aₓ·xᵐ·yⁿ with respect to x, the term yⁿ is treated as a constant multiplier because y is held fixed. This reduces the problem to differentiating aₓ·(yⁿ)·xᵐ with respect to x alone, which is a standard single-variable power rule: d/dx [C·xᵐ] = C·m·x^(m−1). The multivariable nature of the function does not change the mechanics — it just changes what you treat as a constant. This is why all single-variable differentiation rules (power, product, chain) extend naturally to partial differentiation.