Chapter 7 微分 - 公式表

\[f'(x) > 0 \Rightarrow f(x) \text{ 单调递增}\] \[f'(x) < 0 \Rightarrow f(x) \text{ 单调递减}\] \[f'(x) = 0 \Rightarrow f(x) \text{ 可能有极值}\]

\[f'(x) \geq 0 \text{ 且 } f'(x) \text{ 不恒为0} \Rightarrow f(x) \text{ 严格递增}\] \[f'(x) \leq 0 \text{ 且 } f'(x) \text{ 不恒为0} \Rightarrow f(x) \text{ 严格递减}\]

\[f'(x) = 0 \text{ 的点称为驻点}\]

\[\text{若 } f'(x_0) = 0 \text{ 且 } f''(x_0) > 0 \Rightarrow x_0 \text{ 是极小值点}\] \[\text{若 } f'(x_0) = 0 \text{ 且 } f''(x_0) < 0 \Rightarrow x_0 \text{ 是极大值点}\] \[\text{若 } f'(x_0) = 0 \text{ 且 } f''(x_0) = 0 \Rightarrow \text{需要进一步判断}\]

\[\text{若 } f'(x) \text{ 在 } x_0 \text{ 左侧为正，右侧为负} \Rightarrow x_0 \text{ 是极大值点}\] \[\text{若 } f'(x) \text{ 在 } x_0 \text{ 左侧为负，右侧为正} \Rightarrow x_0 \text{ 是极小值点}\]

\[\text{若 } f(x) \text{ 在 } x_0 \text{ 处有极值，则 } f'(x_0) = 0\] \[\text{若 } f(x) \text{ 在 } x_0 \text{ 处有拐点，则 } f''(x_0) = 0\]

\[\text{原函数递增} \Leftrightarrow \text{梯度函数为正}\] \[\text{原函数递减} \Leftrightarrow \text{梯度函数为负}\] \[\text{原函数有极值} \Leftrightarrow \text{梯度函数为零}\]

\[\text{求函数 } f(x) \text{ 的最大值或最小值}\] \[\text{步骤：1. 求导 2. 令导数为零 3. 判断极值}\]

\[\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}\]

\[\text{面积最大：} A = xy \text{，约束条件 } g(x,y) = 0\] \[\text{体积最大：} V = xyz \text{，约束条件 } h(x,y,z) = 0\]

\[\frac{d}{dx}(x^n) = nx^{n-1}\] \[\frac{d}{dx}(x) = 1\] \[\frac{d}{dx}(c) = 0\]

\[\frac{d}{dx}(e^x) = e^x\] \[\frac{d}{dx}(a^x) = a^x \ln a\] \[\frac{d}{dx}(\ln x) = \frac{1}{x}\]

\[\frac{d}{dx}(\sin x) = \cos x\] \[\frac{d}{dx}(\cos x) = -\sin x\] \[\frac{d}{dx}(\tan x) = \sec^2 x\]

\[\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)\] \[\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)\] \[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}\]