微积分基本公式汇总

微积分基本公式汇总

1. 若 \(f(x) = x^n\)，有 \(f'(x) = nx^{n - 1}\).
2. \(\cfrac{d}{dx}(g(x) + h(x)) = \cfrac{dg}{dx} + \cfrac{dh}{dx}\).
3. \(\cfrac{d}{dx}(g(x)h(x)) = g(x)h'(x) + h(x)g'(x)\)（左乘右导，右乘左导）.
4. \(\cfrac{d}{dx}(g(h(x))) = \cfrac{dg}{dh}(h(x))\cfrac{dh}{dx}(x)\).
5. 若 \(f(x) = n^x\)，有 \(f'(x) = \ln(n) \cdot n^x\).
6. 洛必达法则：\(\displaystyle\lim_{x \to \infty} \cfrac{f(x)}{F(x)} = \lim_{x \to \infty} \cfrac{f'(x)}{F'(x)}\).
7. 微积分基本定理：\(\displaystyle\int_{a}^b f(x) \cdot dx = F(b) - F(a)\).
8. 函数 \(f(x)\) 在 \(x = x_0\) 处的泰勒级数为 \(\displaystyle \sum_{n = 0}^{\infty} \cfrac{f^{(n)}(x_0)}{n!} (x - x_0)^n\).