Comprehensive Differentiation Formulas & Rules Dataset
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.
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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.
Showing 57 of 57
| Category | Function f(x) | Derivative f'(x) | Condition | Notes |
|---|---|---|---|---|
| Basic Rules | c (constant) | 0 | - | Constant rule |
| Basic Rules | x | 1 | - | Identity function |
| Basic Rules | c·f(x) | c·f'(x) | c is constant | Constant multiple rule |
| Basic Rules | f(x) + g(x) | f'(x) + g'(x) | - | Sum rule |
| Basic Rules | f(x) - g(x) | f'(x) - g'(x) | - | Difference rule |
| Basic Rules | f(x)·g(x) | f'(x)g(x) + f(x)g'(x) | - | Product rule |
| Basic Rules | f(x)/g(x) | [f'(x)g(x) - f(x)g'(x)]/[g(x)]² | g(x) ≠ 0 | Quotient rule |
| Basic Rules | f(g(x)) | f'(g(x))·g'(x) | - | Chain rule |
| Power | x^n | n·x^(n-1) | - | Power rule |
| Power | 1/x | -1/x² | x ≠ 0 | Same as x^(-1) |
| Power | 1/x^n | -n/x^(n+1) | x ≠ 0 | Negative power |
| Power | √x | 1/(2√x) | x > 0 | Square root |
| Power | ∜x (x^(1/n)) | 1/(n·x^((n-1)/n)) | x > 0 | nth root |
| Power | x^x | x^x(ln(x) + 1) | x > 0 | Logarithmic differentiation |
| Exponential | e^x | e^x | - | Natural exponential |
| Exponential | a^x | a^x·ln(a) | a > 0, a ≠ 1 | General exponential |
| Exponential | e^(f(x)) | e^(f(x))·f'(x) | - | Chain rule applied |
| Exponential | a^(f(x)) | a^(f(x))·ln(a)·f'(x) | a > 0 | General form with chain rule |
| Logarithmic | ln(x) | 1/x | x > 0 | Natural logarithm |
| Logarithmic | log_a(x) | 1/(x·ln(a)) | x > 0, a > 0 | General logarithm |
| Logarithmic | ln(f(x)) | f'(x)/f(x) | f(x) > 0 | Chain rule applied |
| Logarithmic | log_a(f(x)) | f'(x)/(f(x)·ln(a)) | f(x) > 0 | General form with chain rule |
| Logarithmic | ln|x| | 1/x | x ≠ 0 | Absolute value logarithm |
| Trigonometric | sin(x) | cos(x) | - | Sine function |
| Trigonometric | cos(x) | -sin(x) | - | Cosine function |
| Trigonometric | tan(x) | sec²(x) | x ≠ π/2 + nπ | Tangent function |
| Trigonometric | cot(x) | -csc²(x) | x ≠ nπ | Cotangent function |
| Trigonometric | sec(x) | sec(x)tan(x) | x ≠ π/2 + nπ | Secant function |
| Trigonometric | csc(x) | -csc(x)cot(x) | x ≠ nπ | Cosecant function |
| Trigonometric | sin(f(x)) | cos(f(x))·f'(x) | - | Chain rule applied |
| Trigonometric | cos(f(x)) | -sin(f(x))·f'(x) | - | Chain rule applied |
| Trigonometric | tan(f(x)) | sec²(f(x))·f'(x) | - | Chain rule applied |
| Inverse Trig | arcsin(x) | 1/√(1-x²) | |x| < 1 | Arcsine function |
| Inverse Trig | arccos(x) | -1/√(1-x²) | |x| < 1 | Arccosine function |
| Inverse Trig | arctan(x) | 1/(1+x²) | - | Arctangent function |
| Inverse Trig | arccot(x) | -1/(1+x²) | - | Arccotangent function |
| Inverse Trig | arcsec(x) | 1/(|x|√(x²-1)) | |x| > 1 | Arcsecant function |
| Inverse Trig | arccsc(x) | -1/(|x|√(x²-1)) | |x| > 1 | Arccosecant function |
| Inverse Trig | arcsin(f(x)) | f'(x)/√(1-[f(x)]²) | |f(x)| < 1 | Chain rule applied |
| Inverse Trig | arctan(f(x)) | f'(x)/(1+[f(x)]²) | - | Chain rule applied |
| Hyperbolic | sinh(x) | cosh(x) | - | Hyperbolic sine |
| Hyperbolic | cosh(x) | sinh(x) | - | Hyperbolic cosine |
| Hyperbolic | tanh(x) | sech²(x) | - | Hyperbolic tangent |
| Hyperbolic | coth(x) | -csch²(x) | x ≠ 0 | Hyperbolic cotangent |
| Hyperbolic | sech(x) | -sech(x)tanh(x) | - | Hyperbolic secant |
| Hyperbolic | csch(x) | -csch(x)coth(x) | x ≠ 0 | Hyperbolic cosecant |
| Inverse Hyperbolic | arcsinh(x) | 1/√(x²+1) | - | Inverse hyperbolic sine |
| Inverse Hyperbolic | arccosh(x) | 1/√(x²-1) | x > 1 | Inverse hyperbolic cosine |
| Inverse Hyperbolic | arctanh(x) | 1/(1-x²) | |x| < 1 | Inverse hyperbolic tangent |
| Inverse Hyperbolic | arccoth(x) | 1/(1-x²) | |x| > 1 | Inverse hyperbolic cotangent |
| Inverse Hyperbolic | arcsech(x) | -1/(x√(1-x²)) | 0 < x < 1 | Inverse hyperbolic secant |
| Inverse Hyperbolic | arccsch(x) | -1/(|x|√(1+x²)) | x ≠ 0 | Inverse hyperbolic cosecant |
| Special | |x| | x/|x| = sgn(x) | x ≠ 0 | Absolute value |
| Special | [f(x)]^n | n[f(x)]^(n-1)·f'(x) | - | Generalized power rule |
| Special | [f(x)]^g(x) | [f(x)]^g(x)·[g'(x)ln(f(x)) + g(x)f'(x)/f(x)] | f(x) > 0 | Logarithmic differentiation |
| Special | e^(x²) | 2x·e^(x²) | - | Gaussian form |
| Special | ln(ln(x)) | 1/(x·ln(x)) | x > 1 | Nested logarithm |
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.