常见基本初等函数求导公式

1. 常数函数

• 若 $$f(x) = C$$ (C为常数)
• 则 $$f’(x) = 0$$

2. 幂函数

• 若 $$f(x) = x^n$$ (n为实数)
• 则 $$f’(x) = nx^{n-1}$$

3. 指数函数

• 若 $$f(x) = a^x$$ (a > 0, a ≠ 1)
• 则 $$f’(x) = a^x\ln a$$
• 特别地，当 $$f(x) = e^x$$ 时
• 则 $$f’(x) = e^x$$

4. 对数函数

• 若 $$f(x) = \log_a x$$ (a > 0, a ≠ 1)
• 则 $$f’(x) = \frac{1}{x\ln a}$$
• 特别地，当 $$f(x) = \ln x$$ 时
• 则 $$f’(x) = \frac{1}{x}$$

5. 三角函数

• $$(\sin x)’ = \cos x$$
• $$(\cos x)’ = -\sin x$$
• $$(\tan x)’ = \sec^2 x$$
• $$(\cot x)’ = -\csc^2 x$$
• $$(\sec x)’ = \sec x\tan x$$
• $$(\csc x)’ = -\csc x\cot x$$

6. 反三角函数

• $$(\arcsin x)’ = \frac{1}{\sqrt{1-x^2}}$$
• $$(\arccos x)’ = -\frac{1}{\sqrt{1-x^2}}$$
• $$(\arctan x)’ = \frac{1}{1+x^2}$$
• $$(\text{arccot}\ x)’ = -\frac{1}{1+x^2}$$