{
  "id": "implicit-differentiation",
  "version": "0f5fd6a41854",
  "status": "published",
  "name": "Implicit Differentiation Calculator",
  "question": "Find dy/dx by implicit differentiation",
  "summary": "Finds dy/dx for an equation in x and y, such as x^2 + y^2 = 25, and the slope at a point on the curve, checked numerically.",
  "category": "math",
  "subcategory": "calculus",
  "url": "https://www.acalculator.org/math/implicit-differentiation-calculator",
  "markdown": "https://www.acalculator.org/math/implicit-differentiation-calculator.md",
  "kind": "function",
  "method": "Write the equation as H(x, y) = 0 (left side minus right side). Then dy/dx = −(∂H/∂x)/(∂H/∂y), checked against difference quotients of H.",
  "assumptions": [
    "y is a differentiable function of x near each point; angles are in radians; ln is the natural logarithm.",
    "An answer that fails its check is not shown."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "eq": {
        "title": "Equation in x and y",
        "description": "An equation such as x^2 + y^2 = 25 or x^3 sin(y) + y = 4x + 3.",
        "type": "string",
        "maxLength": 200
      },
      "at": {
        "title": "Slope at (x, y) (optional)",
        "description": "A point on the curve, typed like 3, -4.",
        "type": "string",
        "maxLength": 60
      }
    }
  },
  "outputs": {
    "derivative": {
      "label": "dy/dx =",
      "description": "The derivative of y with respect to x, in terms of x and y.",
      "format": "math"
    },
    "slope": {
      "label": "Slope at the point",
      "description": "dy/dx at the given point of the curve.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "eq": "x^2 + y^2 = 25"
    },
    "outputs": {
      "derivative": "-x/y"
    },
    "text": "For x^2 + y^2 = 25, dy/dx = -x/y."
  },
  "examples": [
    {
      "given": {
        "eq": "x^2 + y^2 = 25",
        "at": "3, -4"
      },
      "expect": {
        "derivative": "-x/y",
        "slope": 0.75
      },
      "source": "OpenStax, Calculus Volume 1, section 3.8 Implicit Differentiation, Example 3.68 and 3.71. https://openstax.org/books/calculus-volume-1/pages/3-8-implicit-differentiation"
    },
    {
      "given": {
        "eq": "x^3 sin(y) + y = 4x + 3"
      },
      "expect": {
        "derivative": "(4 - 3x^2 sin(y))/(x^3 cos(y) + 1)"
      },
      "source": "OpenStax, Calculus Volume 1, section 3.8 Implicit Differentiation, Example 3.69"
    },
    {
      "given": {
        "eq": "y^3 + x^3 - 3x y = 0",
        "at": "3/2, 3/2"
      },
      "expect": {
        "derivative": "(y - x^2)/(y^2 - x)",
        "slope": -1
      },
      "source": "OpenStax, Calculus Volume 1, section 3.8 Implicit Differentiation, Example 3.72 ((3y − 3x²)/(3y² − 3x))"
    }
  ],
  "sources": [
    "OpenStax, Calculus Volume 1, section 3.8 Implicit Differentiation: https://openstax.org/books/calculus-volume-1/pages/3-8-implicit-differentiation"
  ],
  "related": [
    "tangent-line",
    "logarithmic-differentiation",
    "instantaneous-rate-of-change",
    "slope"
  ],
  "changelog": []
}