Differentiation (Pure 3) · 微分(Pure 3)
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| derivative/dɪˈrɪvətɪv/ | 导数 | dǎo shù |
| inverse tangent/ɪnˈvɜːs ˈtændʒənt/ | 反正切 | fǎn zhèng qiè |
| chain rule/tʃeɪn ruːl/ | 链式法则 | liàn shì fǎ zé |
| parametric/ˌpærəˈmetrɪk/ | 参数 | cān shù |
| product rule/ˈprɒdʌkt ruːl/ | 乘积法则 | chéng jī fǎ zé |
| reciprocal/rɪˈsɪprəkl/ | 倒数 | dào shǔ |
| implicit differentiation/ɪmˈplɪsɪt ˌdɪfəˌrenʃɪˈeɪʃn/ | 隐函数微分 | yǐn hán shù wēi fēn |
The derivative 导数 that completes the circle
- You know the derivatives of $\sin$, $\cos$, $\tan$, $e^x$, and $\ln x$. Pure 3 adds one more: the inverse tangent 反正切.
- Together, these standard derivatives let you differentiate any combination of elementary functions.
补全圆圈的导数
- 你知道 $\sin$、$\cos$、$\tan$、$e^x$ 和 $\ln x$ 的导数。Pure 3 增加一个:反正切(inverse tangent)。
- 这些标准导数合在一起,让你能微分初等函数的任何组合。
Review of Pure 2 methods
- The methods are those of Pure 2: the product, quotient and chain rules 链式法则, with parametric 参数 and implicit curves.
- These are the workhorses of differentiation — master them and you can handle any problem.
Worked example. $y = x^2 e^{3x}$. Product rule 乘积法则 with $u = x^2$, $v = e^{3x}$: $\dfrac{dy}{dx} = 2x \cdot e^{3x} + x^2 \cdot 3e^{3x} = e^{3x}(2x + 3x^2)$.
z on an Argand diagram: modulus is the distance, argument the angle, conjugate the reflection
回顾 Pure 2 的方法
- 方法就是 Pure 2 的那些:乘积、商和链式法则,加上参数曲线和隐函数曲线。
- 这些是微分的主力——掌握它们,你就能处理任何问题。
算例。 $y = x^2 e^{3x}$。乘积法则,$u = x^2$,$v = e^{3x}$: $\dfrac{dy}{dx} = 2x \cdot e^{3x} + x^2 \cdot 3e^{3x} = e^{3x}(2x + 3x^2)$.
阿尔冈图上的 z:模是距离,辐角是角度,共轭是反射
The gradient at a point · 一个点处的斜率
gradient = dy/dx
Implicit or not, the derivative is still the slope of the tangent · 相切 — slide the point to see it. · 无论隐式与否,导数仍然是切线的斜率——滑动这个点来看它。
Pure 3 differentiation still uses the product, quotient and chain rules from Pure 2. · Pure 3 的微分仍然使用 Pure 2 的乘积、商和链式法则。
Those rules carry over; only the inverse-tangent derivative is new. · 那些法则延续下来;只有反正切的导数是新的。
If y = x²e^(3x), then dy/dx = e^(3x)(2x + 3x²). At x = 0, dy/dx = ? · 如果 y = x²e^(3x),那么 dy/dx = e^(3x)(2x + 3x²)。在 x = 0,dy/dx = ?
At x = 0: e⁰(0 + 0) = 1 × 0 = 0. · 在 x = 0:e⁰(0 + 0) = 1 × 0 = 0。
The inverse tangent derivative
- One new standard derivative:
- This appears in integration (the reverse) and in related rates problems.
$\tan^{-1}x$ is NOT $\dfrac{1}{\tan x}$. The notation $\tan^{-1}x$ means the inverse function (arctan), not the reciprocal 倒数. The reciprocal is $\cot x$.
反正切的导数
- 一个新的标准导数:
- 这出现在积分(它的逆)和相关变化率问题中。
$\tan^{-1}x$ 不是 $\dfrac{1}{\tan x}$。 记号 $\tan^{-1}x$ 表示反函数(arctan),不是倒数。倒数是 $\cot x$。
The derivative of tan⁻¹x is 1/(1+x²). What is its value at x = 1? · tan⁻¹x 的导数是 1/(1+x²)。它在 x = 1 处的值是多少?
1/(1 + 1²) = 1/2 = 0.5. · 1/(1 + 1²) = 1/2 = 0.5。
What is the derivative of tan⁻¹x at x = 0? · tan⁻¹x 在 x = 0 处的导数是多少?
1/(1 + 0²) = 1/1 = 1. · 1/(1 + 0²) = 1/1 = 1。
tan⁻¹x means 1/tan x. · tan⁻¹x 表示 1/tan x。
tan⁻¹x is the inverse function (arctan), not the reciprocal. The reciprocal is cot x. · tan⁻¹x 是反函数(arctan),不是倒数。倒数是 cot x。
Implicit differentiation 隐函数微分 revisited
- For equations like $x^2 + y^2 = 25$, differentiate every term with respect to $x$.
- When you meet a $y$ term, apply the chain rule: $\dfrac{d}{dx}(y^2) = 2y\dfrac{dy}{dx}$.
- Then solve for $\dfrac{dy}{dx}$.
allow_no_figure: Review lesson consolidating Pure 2 differentiation methods (product, quotient, chain rules) with one new derivative — no new visual concepts introduced.
重温隐函数微分
- 对像 $x^2 + y^2 = 25$ 这样的方程,对每一项关于 $x$ 微分。
- 当你遇到一个 $y$ 项时,应用链式法则:$\dfrac{d}{dx}(y^2) = 2y\dfrac{dy}{dx}$。
- 然后解出 $\dfrac{dy}{dx}$。
allow_no_figure: 复习课巩固纯数学2微分方法（乘积法则、商法则、链式法则）并引入一个新导数——未引入新的视觉概念。
Worked example — implicit differentiation
- Find $\dfrac{dy}{dx}$ for $x^2 + xy + y^2 = 7$.
- Differentiate: $2x + y + x\dfrac{dy}{dx} + 2y\dfrac{dy}{dx} = 0$.
- Collect $\dfrac{dy}{dx}$ terms: $(x + 2y)\dfrac{dy}{dx} = -(2x + y)$.
- $\dfrac{dy}{dx} = -\dfrac{2x + y}{x + 2y}$.
- Differentiate relations given implicitly or parametrically.
算例——隐函数微分
- 求 $\dfrac{dy}{dx}$ 对于 $x^2 + xy + y^2 = 7$ 的值。
- 微分:$2x + y + x\dfrac{dy}{dx} + 2y\dfrac{dy}{dx} = 0$。
- 收集 $\dfrac{dy}{dx}$ 项:$(x + 2y)\dfrac{dy}{dx} = -(2x + y)$。
- $\dfrac{dy}{dx} = -\dfrac{2x + y}{x + 2y}$.
- 对隐式(implicitly)或参数式(parametrically)给出的关系求导。
When differentiating y² implicitly with respect to x, you get: · 当对 y² 关于 x 隐式微分时,你得到:
By the chain rule: d/dx(y²) = 2y · dy/dx. · 由链式法则:d/dx(y²) = 2y · dy/dx。
You've got it
- reuse the product / quotient / chain rules from Pure 2
- the new derivative: $\dfrac{d}{dx}\tan^{-1}x = \dfrac{1}{1 + x^2}$
- still handle parametric and implicit curves the same way
你掌握了
- 重用 Pure 2 的乘积 / 商 / 链式法则
- 新的导数:$\dfrac{d}{dx}\tan^{-1}x = \dfrac{1}{1 + x^2}$
- 仍然以同样的方式处理参数和隐函数曲线