Derivatives Table
Complete reference table of basic derivatives organized by category: polynomial, exponential, trigonometric, and differentiation rules.
Polynomial & Power
| f(x) | f′(x) | |
|---|---|---|
| C (const) | 0 | |
| x | 1 | |
| xⁿ | n·xⁿ⁻¹ | |
| √x = x^(1/2) | 1 / (2√x) | |
| 1/x = x⁻¹ | −1/x² |
Exponential & Logarithmic
| f(x) | f′(x) | |
|---|---|---|
| eˣ | eˣ | |
| aˣ | aˣ · ln(a) | |
| ln(x) | 1/x | |
| log_a(x) | 1 / (x · ln(a)) |
Trigonometric
| f(x) | f′(x) | |
|---|---|---|
| sin(x) | cos(x) | |
| cos(x) | −sin(x) | |
| tan(x) | 1/cos²(x) = sec²(x) | |
| cot(x) | −1/sin²(x) = −csc²(x) | |
| arcsin(x) | 1/√(1 − x²) | |
| arccos(x) | −1/√(1 − x²) | |
| arctan(x) | 1/(1 + x²) |
Differentiation Rules
| f(x) | f′(x) | |
|---|---|---|
| f(x) ± g(x) | f′(x) ± g′(x) | Sum / Difference Rule |
| f(x) · g(x) | f′·g + f·g′ | Product Rule |
| f(x) / g(x) | (f′·g − f·g′) / g² | Quotient Rule |
| f(g(x)) | f′(g(x)) · g′(x) | Chain Rule |
How to Use the Derivatives Table
- Browse the reference table to find the values you need.
- Use search or scroll to locate specific entries.
- Click on a value to copy it or see more details.
- Use the table as a quick reference during calculations or study.
快速参考
| 从 | 到 |
|---|---|
| 1 × 1 | 1 |
| 5 × 5 | 25 |
| 7 × 8 | 56 |
| 9 × 9 | 81 |
| 12 × 12 | 144 |
| 15 × 15 | 225 |
使用场景
- •Quick lookup of values during math class or professional work.
- •Verifying calculations without needing a full scientific calculator.
- •Studying mathematical relationships, patterns, and properties.
- •Using as a handy reference during engineering or science tasks.
公式
The derivative f′(x) measures the instantaneous rate of change of f(x). It is defined as the limit: f′(x) = lim[h→0] (f(x+h) − f(x)) / h.
常见问题
What is a derivative?
A derivative measures the rate of change of a function. Geometrically, f′(x) gives the slope of the tangent line to f(x) at point x.
What are the basic differentiation rules?
The key rules are: Sum Rule (f±g)′ = f′±g′, Product Rule (fg)′ = f′g + fg′, Quotient Rule (f/g)′ = (f′g − fg′)/g², and Chain Rule (f(g(x)))′ = f′(g(x))·g′(x).
What is the derivative of eˣ?
The derivative of eˣ is eˣ itself. This unique property makes e (Euler's number ≈ 2.71828) fundamental in calculus.
How do I find the derivative of a composite function?
Use the chain rule: if y = f(g(x)), then dy/dx = f′(g(x)) · g′(x). For example, d/dx[sin(x²)] = cos(x²) · 2x.