{
  "id": "instantaneous-rate-of-change",
  "version": "f632eac5207e",
  "status": "published",
  "name": "Instantaneous Rate of Change Calculator",
  "question": "Find the instantaneous rate of change",
  "summary": "Finds the instantaneous rate of change f′(a) of a function at x = a, exactly and as a decimal, checked numerically.",
  "category": "math",
  "subcategory": "calculus",
  "url": "https://www.acalculator.org/math/instantaneous-rate-of-change-calculator",
  "markdown": "https://www.acalculator.org/math/instantaneous-rate-of-change-calculator.md",
  "kind": "function",
  "method": "The rate of change at a is the derivative f′(a) = lim (f(a + h) − f(a))/h as h → 0. A computer algebra system finds it; it is shown only when it matches difference quotients.",
  "assumptions": [
    "The variable is x; 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": {
      "f": {
        "title": "Function f(x)",
        "description": "The quantity as a function of x, typed like 3x^2 - 4x + 1 or sin(x).",
        "type": "string",
        "maxLength": 200
      },
      "a": {
        "title": "At x = a",
        "description": "Where the rate is taken: a number or a constant such as pi/2.",
        "type": "string",
        "maxLength": 40
      }
    }
  },
  "outputs": {
    "rate": {
      "label": "Rate of change f′(a)",
      "description": "The derivative of f at a, written exactly.",
      "format": "math"
    },
    "value": {
      "label": "As a decimal",
      "description": "f′(a) to 10 significant figures.",
      "format": "number"
    },
    "derivative": {
      "label": "Derivative f′(x)",
      "description": "The derivative of f at every x.",
      "format": "math"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "f": "3x^2 - 4x + 1",
      "a": "2"
    },
    "outputs": {
      "rate": "8",
      "value": 8,
      "derivative": "6x - 4"
    },
    "text": "The instantaneous rate of change of 3x^2 - 4x + 1 at x = 2 is 8."
  },
  "examples": [
    {
      "given": {
        "f": "3x^2 - 4x + 1",
        "a": "2"
      },
      "expect": {
        "rate": "8",
        "value": 8,
        "derivative": "6x - 4"
      },
      "source": "OpenStax, Calculus Volume 1, section 3.1 Defining the Derivative, Example 3.5. https://openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative"
    },
    {
      "given": {
        "f": "0.4x^2 - 4x + 70",
        "a": "3"
      },
      "expect": {
        "rate": "-8/5",
        "value": -1.6
      },
      "source": "OpenStax, Calculus Volume 1, section 3.1 Defining the Derivative, Example 3.9 (temperature T(t) = 0.4t² − 4t + 70 at t = 3: −1.6 °F an hour)"
    }
  ],
  "sources": [
    "OpenStax, Calculus Volume 1, section 3.1 Defining the Derivative: https://openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative"
  ],
  "related": [
    "tangent-line",
    "linear-approximation",
    "integral",
    "slope"
  ],
  "changelog": []
}