All Derivative Formulas
Complete reference guide with all derivative formulas organized by category. Includes power rule, product rule, quotient rule, chain rule, trigonometric derivatives, exponential derivatives, logarithmic derivatives, and more. Bookmark this page for quick access to differentiation formulas during homework and exams.
This page contains every derivative formula you'll need for calculus. Use the table of contents to jump to specific categories, or scroll through for a complete overview. Each formula is ready to copy and use!
📑 Formula Categories
📐 Basic Derivative Rules
These fundamental derivative formulas are the foundation of differential calculus.
| Rule Name | Formula | Example |
|---|---|---|
| Constant Rule | d/dx[c] = 0 | d/dx[5] = 0 |
| Power Rule | d/dx[x^n] = n·x^(n-1) | d/dx[x³] = 3x² |
| Constant Multiple | d/dx[c·f] = c·f' | d/dx[5x²] = 10x |
| Sum Rule | d/dx[f + g] = f' + g' | d/dx[x² + x³] = 2x + 3x² |
| Difference Rule | d/dx[f - g] = f' - g' | d/dx[x³ - 2x] = 3x² - 2 |
The power rule works for all real numbers n, including negative and fractional exponents! For example: d/dx[x^(-2)] = -2x^(-3) and d/dx[√x] = d/dx[x^(1/2)] = (1/2)x^(-1/2)
⚡ Advanced Derivative Rules
These rules handle more complex scenarios involving products, quotients, and composite functions.
| Rule Name | Formula | Example |
|---|---|---|
| Product Rule | d/dx[f·g] = f'·g + f·g' | d/dx[x²·sin(x)] = 2x·sin(x) + x²·cos(x) |
| Quotient Rule | d/dx[f/g] = (f'·g - f·g')/g² | d/dx[x/sin(x)] = (sin(x) - x·cos(x))/sin²(x) |
| Chain Rule | d/dx[f(g(x))] = f'(g(x))·g'(x) | d/dx[sin(x²)] = cos(x²)·2x |
- Product Rule: d/dx[f·g] ≠ f'·g' (You MUST use f'g + fg')
- Quotient Rule: Remember the order! It's (low d-high minus high d-low) / low²
- Chain Rule: Don't forget to multiply by the inner derivative!
Learn more about these rules in our detailed guides: Product Rule Tutorial, Chain Rule Guide
📊 Trigonometric Derivatives
Trigonometric function derivatives are essential for calculus and physics applications.
| Function | Derivative | Notes |
|---|---|---|
| sin(x) | cos(x) | Most common trig derivative |
| cos(x) | -sin(x) | Note the negative sign! |
| tan(x) | sec²(x) | Also equals 1 + tan²(x) |
| cot(x) | -csc²(x) | Negative of csc²(x) |
| sec(x) | sec(x)·tan(x) | Product of function and tan |
| csc(x) | -csc(x)·cot(x) | Negative product |
For co-functions (cos, cot, csc), the derivative has a negative sign. Regular trig functions (sin, tan, sec) have positive derivatives!
🔢 Exponential & Logarithmic Derivatives
Exponential and logarithmic derivatives are crucial for growth, decay, and optimization problems.
| Function | Derivative | Notes |
|---|---|---|
| e^x | e^x | Derivative equals itself! |
| a^x | a^x·ln(a) | For any constant base a > 0 |
| ln(x) | 1/x | Natural logarithm |
| log_a(x) | 1/(x·ln(a)) | Logarithm base a |
| ln|x| | 1/x | Works for x < 0 too |
The function e^x is unique because it's the only function that equals its own derivative! This makes it incredibly important in differential equations.
🔄 Inverse Trigonometric Derivatives
Inverse trig derivatives appear in integration and arc length problems.
| Function | Derivative | Domain |
|---|---|---|
| arcsin(x) or sin⁻¹(x) | 1/√(1 - x²) | -1 < x < 1 |
| arccos(x) or cos⁻¹(x) | -1/√(1 - x²) | -1 < x < 1 |
| arctan(x) or tan⁻¹(x) | 1/(1 + x²) | All real x |
| arccot(x) or cot⁻¹(x) | -1/(1 + x²) | All real x |
| arcsec(x) or sec⁻¹(x) | 1/(|x|√(x² - 1)) | |x| > 1 |
| arccsc(x) or csc⁻¹(x) | -1/(|x|√(x² - 1)) | |x| > 1 |
🌊 Hyperbolic Function Derivatives
Hyperbolic derivatives are used in engineering, physics, and special relativity.
| Function | Derivative | Notes |
|---|---|---|
| sinh(x) | cosh(x) | Similar to sin → cos |
| cosh(x) | sinh(x) | NO negative sign! |
| tanh(x) | sech²(x) | Like tan → sec² |
| coth(x) | -csch²(x) | Negative version |
| sech(x) | -sech(x)·tanh(x) | Negative product |
| csch(x) | -csch(x)·coth(x) | Negative product |
Notice how hyperbolic derivatives are similar to trig derivatives, but with key differences. The derivative of cosh(x) is sinh(x) with NO negative sign, unlike cos(x) → -sin(x)!
🧮 Practice with Our Calculators
Test these formulas with our free derivative calculators featuring step-by-step solutions!
Try Derivative Calculator →📚 Quick Reference Summary
Master these first:
- Power Rule: d/dx[x^n] = n·x^(n-1)
- Product Rule: (uv)' = u'v + uv'
- Chain Rule: d/dx[f(g(x))] = f'(g(x))·g'(x)
- sin(x): d/dx[sin(x)] = cos(x)
- cos(x): d/dx[cos(x)] = -sin(x)
- e^x: d/dx[e^x] = e^x
- ln(x): d/dx[ln(x)] = 1/x
Deepen your understanding with detailed tutorials: