Calculating Higher-Order Derivatives · 计算高阶导数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| second derivative/ˈsekənd dɪˈrɪvətɪv/ | 二阶导数 | èr jiē dǎo shù |
| higher-order derivatives/ˈhaɪə ˈɔːdə dɪˈrɪvətɪvz/ | 高阶导数 | gāo jiē dǎo shù |
| acceleration/əkˌseləˈreɪʃn/ | 加速度 | jiā sù dù |
Differentiate the derivative
- Nothing stops you from differentiating $f'$ again. The result is the second derivative 二阶导数 $f''$.
- Keep going and you get higher-order derivatives 高阶导数: $f'''$, $f^{(4)}$, and so on.
- Each one measures the rate of change of the one before it.
- The second derivative is the star: it describes how the slope itself is changing.
对导数再求导
- 没有什么能阻止你对 $f'$ 再次求导。结果就是二阶导数 $f''$。
- 继续下去,你得到高阶导数:$f'''$、$f^{(4)}$,等等。
- 每一个都度量前一个的变化率。
- 二阶导数是主角:它描述斜率本身如何变化。
A cubic and its slopes · 三次函数及其斜率
y = ax³ + bx
$f'$ of a cubic is a parabola and $f''$ is a line — each derivative is one degree lower. · 三次函数的 $f'$ 是抛物线，$f''$ 是直线——每阶导数次数降低一次。
Just repeat the process
- Find $f'$ using any rules you need, then differentiate that to get $f''$.
- $f(x)=x^4 \Rightarrow f'(x)=4x^3 \Rightarrow f''(x)=12x^2 \Rightarrow f'''(x)=24x$.
- Each step is an ordinary derivative — no new technique, just applied again.
- Simplify $f'$ before differentiating again; it keeps the algebra clean.
只需重复这个过程
- 用任何需要的规则求 $f'$,再对它求导得到 $f''$。
- $f(x)=x^4 \Rightarrow f'(x)=4x^3 \Rightarrow f''(x)=12x^2 \Rightarrow f'''(x)=24x$。
- 每一步都是普通求导——没有新技巧,只是再用一次。
- 再次求导前先化简 $f'$;这能让代数保持整洁。
For · 支持 $f(x)=x^4$, what is $f'(x)$? · 对于$f(x)=x^4$，$f'(x)$是什么？
$f'=4x^3$, then $f''=12x^2$. · $f'=4x^3$，然后是 $f''=12x^2$。
For · 支持 $f(x)=x^3-2x^2+5x$, $f'(x)=6x-4$. Find $f'(2)$. · 对于 $f(x)=x^3-2x^2+5x$，已知 $f'(x)=6x-4$。求 $f'(2)$。
$f'(2)=6(2)-4=8$.
For · 支持 $f(x)=x^3-2x^2+5x$, select all · 所有 correct derivatives. · 对于 $f(x)=x^3-2x^2+5x$，选择所有正确的导数。
Differentiate step by step; the fourth option is a wrong simplification. · 逐步求导；第四个选项是错误的化简。
Notation for the higher orders
- Prime notation: $f''(x)$, $f'''(x)$, then $f^{(4)}(x)$ (numbers past three primes).
- Leibniz notation: $\dfrac{d^2y}{dx^2}$ for the second derivative, $\dfrac{d^3y}{dx^3}$ for the third.
- Read $\dfrac{d^2y}{dx^2}$ as "d-squared-y d-x-squared" — it means "differentiate $y$ twice."
- The two notations mean exactly the same thing.
高阶的记号
- 撇号记号: $f''(x)$、$f'''(x)$,然后 $f^{(4)}(x)$(超过三撇就用数字)。
- 莱布尼茨记号: 二阶导数用 $\dfrac{d^2y}{dx^2}$,三阶用 $\dfrac{d^3y}{dx^3}$。
- 把 $\dfrac{d^2y}{dx^2}$ 读作"d 平方 y,d x 平方"——意思是"对 $y$ 求导两次"。
- 两种记号意思完全相同。
In Leibniz notation the second derivative of $y$ is written $\dfrac{d^2 y}{dx^{\square}}$. The power $\square$ is . · 在莱布尼茨记法中，$y$ 的二阶导数写作 $\dfrac{d^2 y}{dx^{\square}}$。指数 $\square$ 是。
$\dfrac{d^2y}{dx^2}$.
What the second derivative means
- $f'$ is the rate of change of $f$; $f''$ is the rate of change of $f'$.
- If $f$ is position, then $f'$ is velocity and $f''$ is acceleration — how fast the velocity changes.
- The sign of $f''$ tells you about concavity (curving up or down), coming up in Unit 5.
- So higher-order derivatives aren't busywork — $f''$ carries real, physical meaning.
二阶导数意味着什么
- $f'$ 是 $f$ 的变化率;$f''$ 是 $f'$ 的变化率。
- 若 $f$ 是位置,则 $f'$ 是速度,$f''$ 是加速度——速度变化的快慢。
- $f''$ 的符号告诉你凹凸性(向上还是向下弯),第 5 单元会讲。
- 所以高阶导数不是无用功——$f''$ 承载真实的物理意义。
The second derivative $f'$ equals $(f')^2$. · 二阶导数 $f'$ 等于 $(f')^2$。
$f'$ is the derivative of $f'$, not its square. · $f'$ 是 $f'$ 的导数，而不是它的平方。
If $f$ is position, the second derivative $f'$ represents... · 如果 $f$ 是位置，那么二阶导数 $f'$ 表示...
$f'=$ velocity, $f''=$ acceleration. · $f'=$ 是速度，$f''=$ 是加速度。
$f''$ is the derivative of $f'$, not the square of $f'$: $f''\neq(f')^2$. And apply the chain/product/quotient rules again at each stage — differentiating $f'=\sin(x^2)$ to get $f''$ still needs the chain rule. Don't switch off the rules just because you're on the second round.
$f''$ 是 $f'$ 的导数,而不是 $f'$ 的平方:$f''\neq(f')^2$。而且每一阶都要再次套用链式/乘积/商法则——对 $f'=\sin(x^2)$ 求导得 $f''$ 仍需链式法则。别因为到了第二轮就把规则关掉。
Find $f''(x)$ for $f(x)=x^3-2x^2+5x$.
- First derivative: $f'(x)=3x^2-4x+5$.
- Differentiate again: $f''(x)=6x-4$.
- (One more: $f'''(x)=6$, and $f^{(4)}(x)=0$.)
求 $f(x)=x^3-2x^2+5x$ 的 $f''(x)$。
- 一阶导数:$f'(x)=3x^2-4x+5$。
- 再求导:$f''(x)=6x-4$。
- (再来一次:$f'''(x)=6$,$f^{(4)}(x)=0$。)
A higher-order derivative is found by differentiating repeatedly: $f''$ is the derivative of $f'$, and so on. Write them $f''(x)$ / $\frac{d^2y}{dx^2}$, etc. The second derivative is the rate of change of the first derivative (e.g. acceleration), and its sign will describe concavity. Keep applying the chain/product/quotient rules at every stage.
高阶导数通过反复求导得到:$f''$ 是 $f'$ 的导数,依此类推。记作 $f''(x)$ / $\frac{d^2y}{dx^2}$ 等。二阶导数是一阶导数的变化率(如加速度),其符号将描述凹凸性。每一阶都要继续套用链式/乘积/商法则。