{
  "id": "math-derivative-formulas",
  "name": "Comprehensive Differentiation Formulas & Rules Dataset",
  "category": "math",
  "source": null,
  "updated": "2025-12-28",
  "columns": [
    {
      "key": "category",
      "label": "Category",
      "type": "string"
    },
    {
      "key": "function",
      "label": "Function f(x)",
      "type": "string"
    },
    {
      "key": "derivative",
      "label": "Derivative f'(x)",
      "type": "string"
    },
    {
      "key": "condition",
      "label": "Condition",
      "type": "string"
    },
    {
      "key": "notes",
      "label": "Notes",
      "type": "string"
    }
  ],
  "rows": [
    {
      "category": "Basic Rules",
      "function": "c (constant)",
      "derivative": "0",
      "condition": "-",
      "notes": "Constant rule"
    },
    {
      "category": "Basic Rules",
      "function": "x",
      "derivative": "1",
      "condition": "-",
      "notes": "Identity function"
    },
    {
      "category": "Basic Rules",
      "function": "c·f(x)",
      "derivative": "c·f'(x)",
      "condition": "c is constant",
      "notes": "Constant multiple rule"
    },
    {
      "category": "Basic Rules",
      "function": "f(x) + g(x)",
      "derivative": "f'(x) + g'(x)",
      "condition": "-",
      "notes": "Sum rule"
    },
    {
      "category": "Basic Rules",
      "function": "f(x) - g(x)",
      "derivative": "f'(x) - g'(x)",
      "condition": "-",
      "notes": "Difference rule"
    },
    {
      "category": "Basic Rules",
      "function": "f(x)·g(x)",
      "derivative": "f'(x)g(x) + f(x)g'(x)",
      "condition": "-",
      "notes": "Product rule"
    },
    {
      "category": "Basic Rules",
      "function": "f(x)/g(x)",
      "derivative": "[f'(x)g(x) - f(x)g'(x)]/[g(x)]²",
      "condition": "g(x) ≠ 0",
      "notes": "Quotient rule"
    },
    {
      "category": "Basic Rules",
      "function": "f(g(x))",
      "derivative": "f'(g(x))·g'(x)",
      "condition": "-",
      "notes": "Chain rule"
    },
    {
      "category": "Power",
      "function": "x^n",
      "derivative": "n·x^(n-1)",
      "condition": "-",
      "notes": "Power rule"
    },
    {
      "category": "Power",
      "function": "1/x",
      "derivative": "-1/x²",
      "condition": "x ≠ 0",
      "notes": "Same as x^(-1)"
    },
    {
      "category": "Power",
      "function": "1/x^n",
      "derivative": "-n/x^(n+1)",
      "condition": "x ≠ 0",
      "notes": "Negative power"
    },
    {
      "category": "Power",
      "function": "√x",
      "derivative": "1/(2√x)",
      "condition": "x > 0",
      "notes": "Square root"
    },
    {
      "category": "Power",
      "function": "∜x (x^(1/n))",
      "derivative": "1/(n·x^((n-1)/n))",
      "condition": "x > 0",
      "notes": "nth root"
    },
    {
      "category": "Power",
      "function": "x^x",
      "derivative": "x^x(ln(x) + 1)",
      "condition": "x > 0",
      "notes": "Logarithmic differentiation"
    },
    {
      "category": "Exponential",
      "function": "e^x",
      "derivative": "e^x",
      "condition": "-",
      "notes": "Natural exponential"
    },
    {
      "category": "Exponential",
      "function": "a^x",
      "derivative": "a^x·ln(a)",
      "condition": "a > 0, a ≠ 1",
      "notes": "General exponential"
    },
    {
      "category": "Exponential",
      "function": "e^(f(x))",
      "derivative": "e^(f(x))·f'(x)",
      "condition": "-",
      "notes": "Chain rule applied"
    },
    {
      "category": "Exponential",
      "function": "a^(f(x))",
      "derivative": "a^(f(x))·ln(a)·f'(x)",
      "condition": "a > 0",
      "notes": "General form with chain rule"
    },
    {
      "category": "Logarithmic",
      "function": "ln(x)",
      "derivative": "1/x",
      "condition": "x > 0",
      "notes": "Natural logarithm"
    },
    {
      "category": "Logarithmic",
      "function": "log_a(x)",
      "derivative": "1/(x·ln(a))",
      "condition": "x > 0, a > 0",
      "notes": "General logarithm"
    },
    {
      "category": "Logarithmic",
      "function": "ln(f(x))",
      "derivative": "f'(x)/f(x)",
      "condition": "f(x) > 0",
      "notes": "Chain rule applied"
    },
    {
      "category": "Logarithmic",
      "function": "log_a(f(x))",
      "derivative": "f'(x)/(f(x)·ln(a))",
      "condition": "f(x) > 0",
      "notes": "General form with chain rule"
    },
    {
      "category": "Logarithmic",
      "function": "ln|x|",
      "derivative": "1/x",
      "condition": "x ≠ 0",
      "notes": "Absolute value logarithm"
    },
    {
      "category": "Trigonometric",
      "function": "sin(x)",
      "derivative": "cos(x)",
      "condition": "-",
      "notes": "Sine function"
    },
    {
      "category": "Trigonometric",
      "function": "cos(x)",
      "derivative": "-sin(x)",
      "condition": "-",
      "notes": "Cosine function"
    },
    {
      "category": "Trigonometric",
      "function": "tan(x)",
      "derivative": "sec²(x)",
      "condition": "x ≠ π/2 + nπ",
      "notes": "Tangent function"
    },
    {
      "category": "Trigonometric",
      "function": "cot(x)",
      "derivative": "-csc²(x)",
      "condition": "x ≠ nπ",
      "notes": "Cotangent function"
    },
    {
      "category": "Trigonometric",
      "function": "sec(x)",
      "derivative": "sec(x)tan(x)",
      "condition": "x ≠ π/2 + nπ",
      "notes": "Secant function"
    },
    {
      "category": "Trigonometric",
      "function": "csc(x)",
      "derivative": "-csc(x)cot(x)",
      "condition": "x ≠ nπ",
      "notes": "Cosecant function"
    },
    {
      "category": "Trigonometric",
      "function": "sin(f(x))",
      "derivative": "cos(f(x))·f'(x)",
      "condition": "-",
      "notes": "Chain rule applied"
    },
    {
      "category": "Trigonometric",
      "function": "cos(f(x))",
      "derivative": "-sin(f(x))·f'(x)",
      "condition": "-",
      "notes": "Chain rule applied"
    },
    {
      "category": "Trigonometric",
      "function": "tan(f(x))",
      "derivative": "sec²(f(x))·f'(x)",
      "condition": "-",
      "notes": "Chain rule applied"
    },
    {
      "category": "Inverse Trig",
      "function": "arcsin(x)",
      "derivative": "1/√(1-x²)",
      "condition": "|x| < 1",
      "notes": "Arcsine function"
    },
    {
      "category": "Inverse Trig",
      "function": "arccos(x)",
      "derivative": "-1/√(1-x²)",
      "condition": "|x| < 1",
      "notes": "Arccosine function"
    },
    {
      "category": "Inverse Trig",
      "function": "arctan(x)",
      "derivative": "1/(1+x²)",
      "condition": "-",
      "notes": "Arctangent function"
    },
    {
      "category": "Inverse Trig",
      "function": "arccot(x)",
      "derivative": "-1/(1+x²)",
      "condition": "-",
      "notes": "Arccotangent function"
    },
    {
      "category": "Inverse Trig",
      "function": "arcsec(x)",
      "derivative": "1/(|x|√(x²-1))",
      "condition": "|x| > 1",
      "notes": "Arcsecant function"
    },
    {
      "category": "Inverse Trig",
      "function": "arccsc(x)",
      "derivative": "-1/(|x|√(x²-1))",
      "condition": "|x| > 1",
      "notes": "Arccosecant function"
    },
    {
      "category": "Inverse Trig",
      "function": "arcsin(f(x))",
      "derivative": "f'(x)/√(1-[f(x)]²)",
      "condition": "|f(x)| < 1",
      "notes": "Chain rule applied"
    },
    {
      "category": "Inverse Trig",
      "function": "arctan(f(x))",
      "derivative": "f'(x)/(1+[f(x)]²)",
      "condition": "-",
      "notes": "Chain rule applied"
    },
    {
      "category": "Hyperbolic",
      "function": "sinh(x)",
      "derivative": "cosh(x)",
      "condition": "-",
      "notes": "Hyperbolic sine"
    },
    {
      "category": "Hyperbolic",
      "function": "cosh(x)",
      "derivative": "sinh(x)",
      "condition": "-",
      "notes": "Hyperbolic cosine"
    },
    {
      "category": "Hyperbolic",
      "function": "tanh(x)",
      "derivative": "sech²(x)",
      "condition": "-",
      "notes": "Hyperbolic tangent"
    },
    {
      "category": "Hyperbolic",
      "function": "coth(x)",
      "derivative": "-csch²(x)",
      "condition": "x ≠ 0",
      "notes": "Hyperbolic cotangent"
    },
    {
      "category": "Hyperbolic",
      "function": "sech(x)",
      "derivative": "-sech(x)tanh(x)",
      "condition": "-",
      "notes": "Hyperbolic secant"
    },
    {
      "category": "Hyperbolic",
      "function": "csch(x)",
      "derivative": "-csch(x)coth(x)",
      "condition": "x ≠ 0",
      "notes": "Hyperbolic cosecant"
    },
    {
      "category": "Inverse Hyperbolic",
      "function": "arcsinh(x)",
      "derivative": "1/√(x²+1)",
      "condition": "-",
      "notes": "Inverse hyperbolic sine"
    },
    {
      "category": "Inverse Hyperbolic",
      "function": "arccosh(x)",
      "derivative": "1/√(x²-1)",
      "condition": "x > 1",
      "notes": "Inverse hyperbolic cosine"
    },
    {
      "category": "Inverse Hyperbolic",
      "function": "arctanh(x)",
      "derivative": "1/(1-x²)",
      "condition": "|x| < 1",
      "notes": "Inverse hyperbolic tangent"
    },
    {
      "category": "Inverse Hyperbolic",
      "function": "arccoth(x)",
      "derivative": "1/(1-x²)",
      "condition": "|x| > 1",
      "notes": "Inverse hyperbolic cotangent"
    },
    {
      "category": "Inverse Hyperbolic",
      "function": "arcsech(x)",
      "derivative": "-1/(x√(1-x²))",
      "condition": "0 < x < 1",
      "notes": "Inverse hyperbolic secant"
    },
    {
      "category": "Inverse Hyperbolic",
      "function": "arccsch(x)",
      "derivative": "-1/(|x|√(1+x²))",
      "condition": "x ≠ 0",
      "notes": "Inverse hyperbolic cosecant"
    },
    {
      "category": "Special",
      "function": "|x|",
      "derivative": "x/|x| = sgn(x)",
      "condition": "x ≠ 0",
      "notes": "Absolute value"
    },
    {
      "category": "Special",
      "function": "[f(x)]^n",
      "derivative": "n[f(x)]^(n-1)·f'(x)",
      "condition": "-",
      "notes": "Generalized power rule"
    },
    {
      "category": "Special",
      "function": "[f(x)]^g(x)",
      "derivative": "[f(x)]^g(x)·[g'(x)ln(f(x)) + g(x)f'(x)/f(x)]",
      "condition": "f(x) > 0",
      "notes": "Logarithmic differentiation"
    },
    {
      "category": "Special",
      "function": "e^(x²)",
      "derivative": "2x·e^(x²)",
      "condition": "-",
      "notes": "Gaussian form"
    },
    {
      "category": "Special",
      "function": "ln(ln(x))",
      "derivative": "1/(x·ln(x))",
      "condition": "x > 1",
      "notes": "Nested logarithm"
    }
  ],
  "meta": {
    "seo_title": "Differentiation Formulas - Free Download (Excel/CSV/PDF)",
    "seo_description": "Download 57 common differentiation formulas and derivative rules for calculus. Free in Excel, CSV, & PDF formats for students & professionals.",
    "title": "Comprehensive Differentiation Formulas & Rules Dataset",
    "description": "This dataset provides a comprehensive collection of 57 common differentiation formulas and rules, including basic rules, power rules, trigonometric functions, and more. Each entry lists the function, its derivative, conditions, and notes.",
    "key_takeaways": [
      "Access 57 essential differentiation formulas and derivative rules.",
      "Explore calculus concepts with categorized functions and their derivatives.",
      "Download ready-to-use data for academic study or professional reference.",
      "Leverage detailed conditions and notes for each differentiation rule."
    ],
    "keywords": [
      "differentiation formulas excel",
      "derivative rules csv",
      "calculus formulas pdf",
      "mathematics derivatives list",
      "common differentiation table",
      "free download calculus data"
    ],
    "use_cases": [
      "Import the **CSV** file into your Python scripts or SQL database to build custom calculus learning applications or study tools.",
      "Use the **Excel** file to filter formulas by category, analyze conditions, or create study guides with ease.",
      "Print the **PDF** version for quick offline reference during exams, classroom lectures, or personal study sessions.",
      "Reference this dataset to quickly verify derivatives for complex functions in engineering, physics, or data science calculations."
    ]
  },
  "record_count": 57,
  "title": "Comprehensive Differentiation Formulas & Rules Dataset"
}