求导法则参考
基本法则
| 法则 | 公式 | 示例 |
|---|---|---|
| 常数法则 | d/dx [c] = 0 | d/dx [5] = 0 |
| 幂函数法则 | d/dx [xⁿ] = n·xⁿ⁻¹ | d/dx [x³] = 3x² |
| 常数倍法则 | d/dx [c·f] = c·f' | d/dx [5x²] = 10x |
| 加减法则 | d/dx [f ± g] = f' ± g' | d/dx [x² + x] = 2x + 1 |
| 乘积法则 | d/dx [f·g] = f'·g + f·g' | d/dx [x·sin(x)] = sin(x) + x·cos(x) |
| 商法则 | d/dx [f/g] = (f'g - fg') / g² | d/dx [x/eˣ] = (eˣ - xeˣ) / e²ˣ |
| 链式法则 | d/dx [f(g(x))] = f'(g(x))·g'(x) | d/dx [sin(x²)] = cos(x²)·2x |
常见函数导数
| f(x) | f'(x) |
|---|---|
| xⁿ | n·xⁿ⁻¹ |
| eˣ | eˣ |
| aˣ | aˣ · ln(a) |
| ln(x) | 1/x |
| log_a(x) | 1 / (x · ln(a)) |
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | sec²(x) |
| cot(x) | -csc²(x) |
| sec(x) | sec(x)·tan(x) |
| csc(x) | -csc(x)·cot(x) |
| arcsin(x) | 1 / √(1 - x²) |
| arccos(x) | -1 / √(1 - x²) |
| arctan(x) | 1 / (1 + x²) |
| |x| | x / |x|, x ≠ 0 |