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Partial Derivative Calculator

Enter a two-variable polynomial and choose x or y to get its partial derivative and optional value at a point.

Differentiate a Two-Variable Polynomial

Enter a polynomial in x and y using * between factors, then choose the differentiation variable. This parser does not support trig, logs, roots, quotients, or parentheses.

The polynomial partial derivative and optional evaluated value will appear here.

How to Use Partial Derivative Calculator

1

Enter Supported Polynomial Syntax

Use forms such as 3*x^2*y+4*y^2-5*x with * between factors.

2

Choose x or y

The other variable is treated as a constant during differentiation.

3

Optionally Evaluate a Point

Enter both x and y to evaluate the derivative expression numerically.

Partial Derivative Calculator Features

  • Two-variable polynomial parser
  • Differentiate with respect to x or y
  • Optional point evaluation
  • Unsupported syntax rejected explicitly

Common Use Cases

Check calculus practice

Compare a hand-derived polynomial result with the term-by-term power rule.

Evaluate a rate of change

Enter x and y after differentiation to calculate the partial derivative at one point.

Limits and Important Notes

Result type: Exact calculation

Calculation Details

  • Parse a sum of polynomial terms in x and y.
  • Treat the unselected variable as constant.
  • Apply the power rule to every term.
  • Optionally substitute a complete x-y point.

Formula

For ∂/∂x of a·x^m·y^n, hold y constant and return a·m·x^(m−1)·y^n; apply the analogous rule for y

Rounding

Symbolic coefficients display with up to ten decimals; point values display with up to eight.

Default Assumptions

  • Expressions are sums of x-y polynomial terms.
  • Factors are separated with * and exponents are nonnegative integers from 0 to 20.
  • Only x and y are supported variables.
  • No implicit multiplication, parentheses, trig, logarithms, roots, quotients, or general powers are accepted.

Verified Example

Inputs: f(x,y) = 3*x^2*y + 4*y^2 − 5*x; differentiate with respect to x.

Calculation: 3*x^2*y becomes 6*x*y; 4*y^2 is constant in x; −5*x becomes −5.

Result: ∂f/∂x = 6*x*y−5.

Sources and Version

Version: Two-variable polynomial parser

Last verified: 2026-08-14

Update responsibility: OnlineTool.me

  • This deliberately limited parser is not a general computer algebra system.
  • Equivalent syntax without * may be rejected rather than guessed.
  • The tool does not simplify by combining like terms or verify differentiability outside polynomial rules.
  • For advanced expressions, use a verified CAS and review its domain assumptions.

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Frequently Asked Questions