{
  "id": "partial-derivative",
  "version": "4ce37a9e1b1d",
  "status": "published",
  "name": "Partial Derivative Calculator",
  "question": "What is the partial derivative of f?",
  "summary": "Finds partial derivatives, checked numerically.",
  "category": "math",
  "subcategory": "calculus",
  "url": "https://www.acalculator.org/math/partial-derivative-calculator",
  "markdown": "https://www.acalculator.org/math/partial-derivative-calculator.md",
  "kind": "function",
  "method": "Each step is a CAS derivative checked at 15 points.",
  "assumptions": [
    "Radians."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "f": {
        "title": "Function f(x, y, …)",
        "description": "The function, in one or more letters.",
        "type": "string",
        "maxLength": 200
      },
      "wrt": {
        "title": "With respect to",
        "description": "Letters in order: x for ∂f/∂x, xy for f_xy.",
        "type": "string",
        "maxLength": 4
      },
      "at": {
        "title": "Value at (optional)",
        "description": "A point: x = 1, y = 2 (or 1, 2).",
        "type": "string",
        "maxLength": 60
      }
    }
  },
  "outputs": {
    "derivative": {
      "label": "Partial derivative",
      "description": "The partial derivative.",
      "format": "math"
    },
    "value": {
      "label": "Value at the point",
      "description": "Its value at the point.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "f": "x^2 - 3x y + 2y^2 - 4x + 5y - 12",
      "wrt": "x"
    },
    "outputs": {
      "derivative": "2x - 3y - 4"
    },
    "text": "The partial derivative of x^2 - 3x y + 2y^2 - 4x + 5y - 12 with respect to x is 2x - 3y - 4."
  },
  "examples": [
    {
      "given": {
        "f": "x^2 - 3x y + 2y^2 - 4x + 5y - 12",
        "wrt": "y",
        "at": "1, 2"
      },
      "expect": {
        "derivative": "-3x + 4y + 5",
        "value": 10
      },
      "source": "OpenStax Calculus Vol. 3 (Strang and Herman, 2016), 4.3, Ex. 4.15"
    },
    {
      "given": {
        "f": "x e^(-3y) + sin(2x - 5y)",
        "wrt": "xy"
      },
      "expect": {
        "derivative": "10 sin(2x - 5y) - 3/e^(3y)"
      },
      "source": "OpenStax Calculus Vol. 3 (Strang and Herman, 2016), 4.3, Ex. 4.19"
    }
  ],
  "sources": [
    "OpenStax, Calculus Volume 3, section 4.3 Partial Derivatives: https://openstax.org/books/calculus-volume-3/pages/4-3-partial-derivatives"
  ],
  "related": [
    "derivative",
    "jacobian",
    "curl",
    "lagrange-multiplier"
  ],
  "changelog": []
}