{
  "title": "Derivative walkthrough",
  "human_url": "https://happytails.ai/en/calculators/derivative-walkthrough/",
  "agent_url": "https://happytails.ai/agents/en/calculators/derivative-walkthrough/",
  "language": "en",
  "markdown_url": "https://happytails.ai/en/calculators/derivative-walkthrough.md",
  "content": "Derivative walkthrough\nDifferentiate an expression symbolically and see which rule was used at every step. Numerical spot checks compare the answer with central differences at five points.\nHow can we improve this tool?\nReport a problem or share an idea.\nPlease leave out private information. Your message and the tool name go to our team via PostHog.\nHow to use Derivative walkthrough\nType the expression, such as x^3*ln(x).\nChoose the variable and add assumptions such as x > 0 if you want them used.\nRun the walkthrough and read the rules, the simplified derivative and any conditions.\nCheck the spot checks, then download the JSON report or the CSV of checks.\nUseful next steps\nGraphing calculator\nScientific calculator\nQuadratic formula calculator\nRules and steps\nThe tool applies the constant, sum, product, quotient, power, exponential, logarithm, trigonometric and chain rules to each part of the expression and lists them from the outside in. Letters other than the chosen variable are treated as constants. Method: OpenStax, Calculus Volume 1, chapter 3 Derivatives (2016).\nSimplifying safely\nRewrites that change where an expression is defined, such as x^3/x to x^2 or sqrt(x^2) to x, are only applied when your assumptions allow them, for example x > 0. Otherwise they are listed as gated rewrites and the safe form is kept. Conditions such as x > 0 for ln(x) or x != 0 for x^-1 are listed separately.\nSpot checks are evidence, not proof\nThe derivative is evaluated at five fixed points and compared with a central difference of the original expression. Points where either value is undefined are marked undefined and are not compared. The report counts the compared points and says the checks agree only when at least one point was compared and none disagreed, so sqrt(x - 5), which is undefined at all five points, gets no claim of agreement.",
  "content_format": "text/plain",
  "visibility": "public",
  "links": [
    {
      "name": "Graphing calculator",
      "url": "https://happytails.ai/en/calculators/graphing-calculator/"
    },
    {
      "name": "Scientific calculator",
      "url": "https://happytails.ai/en/calculators/scientific-calculator/"
    },
    {
      "name": "Quadratic formula calculator",
      "url": "https://happytails.ai/en/calculators/quadratic-formula/"
    }
  ],
  "tool": {
    "id": "derivative-walkthrough",
    "category": "calculators",
    "built": true,
    "requirements": "Expressions are limited to 1,000 characters, 512 symbols, 24 nesting levels and 100 digits per number, with a fixed step budget. Supported functions are sin, cos, tan, asin, acos, atan, exp, ln, log, sqrt and abs. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of spot checks.",
    "implementation": {
      "fields": [
        {
          "key": "variable",
          "label": "Differentiate with respect to",
          "value": "x",
          "type": "text",
          "maxLength": 4,
          "help": "One letter other than e, optionally followed by digits. Other letters are treated as constants."
        },
        {
          "key": "assumptions",
          "label": "Assumptions (optional)",
          "value": "",
          "type": "text",
          "maxLength": 200,
          "help": "Optional, such as x > 0 or x != 0, separated by commas. Rewrites like sqrt(x^2) to x are applied only when an assumption allows them."
        }
      ],
      "registrationFamily": "exact-math-reference",
      "inputMaxLength": 1000,
      "inputErrorCode": "EXACT_MATH_REFERENCE_EXPRESSION_TOO_LONG",
      "note": "Differentiates symbolically on your device and checks the result numerically at five points. Expressions are limited to 1,000 characters and 24 nesting levels.",
      "experience": {
        "automatic": false,
        "single": true,
        "inputMaxLength": 1000,
        "inputMode": "text",
        "label": "Expression",
        "placeholder": "x^3*ln(x)",
        "help": "Use x for the variable, ^ for powers and functions such as sin(x) or ln(x). Up to 1,000 characters. Review the settings before running.",
        "action": null
      },
      "mode": "explicit",
      "interactive": false
    },
    "execution": "Browser interface",
    "api": null,
    "browser_agent": {
      "supported": true,
      "name": "run_current_tool",
      "discovery": "WebMCP on the human page in a supporting browser",
      "verification": "See audit/API-AUDIT.md; availability is not verification"
    }
  },
  "markdown": "# Derivative walkthrough\n\n- Human page: https://happytails.ai/en/calculators/derivative-walkthrough/\n- Markdown: https://happytails.ai/en/calculators/derivative-walkthrough.md\n- Structured JSON: https://happytails.ai/agents/en/calculators/derivative-walkthrough/index.json\n- Visibility: public\n\n## Tool capabilities\n\n| Property | Value |\n| --- | --- |\n| Tool ID | `derivative-walkthrough` |\n| Category | calculators |\n| Implementation | Registered implementation. Registration alone is not a production-quality guarantee. |\n| Browser execution | Use the visible action or interactive controls. |\n| Browser processing | Browser; see tool notes for network behavior. |\n| HTTP API | Not callable through HTTP. |\n| Browser-agent interface | run_current_tool |\n\n### Implementation notes\n\nDifferentiates symbolically on your device and checks the result numerically at five points. Expressions are limited to 1,000 characters and 24 nesting levels.\n\n### Inputs and settings\n\nText input: Expression.\n\nExample or placeholder shown in the interface (not a validated fixture):\n\n```text\nx^3*ln(x)\n```\n\nUse x for the variable, ^ for powers and functions such as sin(x) or ln(x). Up to 1,000 characters. Review the settings before running.\n\n| Setting | Meaning | Control type | Default | Constraints |\n| --- | --- | --- | --- | --- |\n| `variable` | Differentiate with respect to | text | \"x\" |  |\n| `assumptions` | Assumptions (optional) | text | \"\" |  |\n\n### Browser-agent workflow\n\n1. Open the human page in a browser that supports WebMCP.\n2. Discover the interface exposed by that page; use its actual schema.\n3. Select files on the page first when the tool requires files.\n4. Call `run_current_tool` with valid input and settings.\n5. Inspect the returned result or error and the displayed output. Retrieve files from the displayed download links.\n\nCalls do not automatically copy or download results. Browser permissions still apply. Interface availability alone is not proof of successful execution.\n\n### Planning specification\n\nExpressions are limited to 1,000 characters, 512 symbols, 24 nesting levels and 100 digits per number, with a fixed step budget. Supported functions are sin, cos, tan, asin, acos, atan, exp, ln, log, sqrt and abs. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of spot checks.\n\nThis describes intended requirements. Use the implementation notes, interface schema and observed output to determine current support.\n\n### Complete machine-readable capability record\n\n```json\n{\n  \"id\": \"derivative-walkthrough\",\n  \"category\": \"calculators\",\n  \"built\": true,\n  \"requirements\": \"Expressions are limited to 1,000 characters, 512 symbols, 24 nesting levels and 100 digits per number, with a fixed step budget. Supported functions are sin, cos, tan, asin, acos, atan, exp, ln, log, sqrt and abs. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of spot checks.\",\n  \"implementation\": {\n    \"fields\": [\n      {\n        \"key\": \"variable\",\n        \"label\": \"Differentiate with respect to\",\n        \"value\": \"x\",\n        \"type\": \"text\",\n        \"maxLength\": 4,\n        \"help\": \"One letter other than e, optionally followed by digits. Other letters are treated as constants.\"\n      },\n      {\n        \"key\": \"assumptions\",\n        \"label\": \"Assumptions (optional)\",\n        \"value\": \"\",\n        \"type\": \"text\",\n        \"maxLength\": 200,\n        \"help\": \"Optional, such as x > 0 or x != 0, separated by commas. Rewrites like sqrt(x^2) to x are applied only when an assumption allows them.\"\n      }\n    ],\n    \"registrationFamily\": \"exact-math-reference\",\n    \"inputMaxLength\": 1000,\n    \"inputErrorCode\": \"EXACT_MATH_REFERENCE_EXPRESSION_TOO_LONG\",\n    \"note\": \"Differentiates symbolically on your device and checks the result numerically at five points. Expressions are limited to 1,000 characters and 24 nesting levels.\",\n    \"experience\": {\n      \"automatic\": false,\n      \"single\": true,\n      \"inputMaxLength\": 1000,\n      \"inputMode\": \"text\",\n      \"label\": \"Expression\",\n      \"placeholder\": \"x^3*ln(x)\",\n      \"help\": \"Use x for the variable, ^ for powers and functions such as sin(x) or ln(x). Up to 1,000 characters. Review the settings before running.\",\n      \"action\": null\n    },\n    \"mode\": \"explicit\",\n    \"interactive\": false\n  },\n  \"execution\": \"Browser interface\",\n  \"api\": null,\n  \"browser_agent\": {\n    \"supported\": true,\n    \"name\": \"run_current_tool\",\n    \"discovery\": \"WebMCP on the human page in a supporting browser\",\n    \"verification\": \"See audit/API-AUDIT.md; availability is not verification\"\n  }\n}\n```\n\n## Page guide\n\nDifferentiate an expression symbolically and see which rule was used at every step. Numerical spot checks compare the answer with central differences at five points.\n\nHow can we improve this tool?\n\nReport a problem or share an idea.\n\nPlease leave out private information. Your message and the tool name go to our team via PostHog.\n\n### How to use Derivative walkthrough\n\n1. Type the expression, such as x^3\\*ln(x).\n2. Choose the variable and add assumptions such as x > 0 if you want them used.\n3. Run the walkthrough and read the rules, the simplified derivative and any conditions.\n4. Check the spot checks, then download the JSON report or the CSV of checks.\n\n## Useful next steps\n\n- [Graphing calculator](https://happytails.ai/en/calculators/graphing-calculator/) ([Markdown](https://happytails.ai/en/calculators/graphing-calculator.md))\n\n- [Scientific calculator](https://happytails.ai/en/calculators/scientific-calculator/) ([Markdown](https://happytails.ai/en/calculators/scientific-calculator.md))\n\n- [Quadratic formula calculator](https://happytails.ai/en/calculators/quadratic-formula/) ([Markdown](https://happytails.ai/en/calculators/quadratic-formula.md))\n\n## Rules and steps\n\nThe tool applies the constant, sum, product, quotient, power, exponential, logarithm, trigonometric and chain rules to each part of the expression and lists them from the outside in. Letters other than the chosen variable are treated as constants. Method: OpenStax, Calculus Volume 1, chapter 3 Derivatives (2016).\n\n### Simplifying safely\n\nRewrites that change where an expression is defined, such as x^3/x to x^2 or sqrt(x^2) to x, are only applied when your assumptions allow them, for example x > 0. Otherwise they are listed as gated rewrites and the safe form is kept. Conditions such as x > 0 for ln(x) or x != 0 for x^-1 are listed separately.\n\n### Spot checks are evidence, not proof\n\nThe derivative is evaluated at five fixed points and compared with a central difference of the original expression. Points where either value is undefined are marked undefined and are not compared. The report counts the compared points and says the checks agree only when at least one point was compared and none disagreed, so sqrt(x - 5), which is undefined at all five points, gets no claim of agreement.\n"
}