Differentiation (Pure 2) · 微分(Pure 2)
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| standard derivatives/ˈstændəd dɪˈrɪvətɪvz/ | 标准导数 | biāo zhǔn dǎo shù |
| quotient rule/ˈkwəʊʃənt ruːl/ | 商法则 | shāng fǎ zé |
| product rule/ˈprɒdʌkt ruːl/ | 乘积法则 | chéng jī fǎ zé |
| parametric/ˌpærəˈmetrɪk/ | 参数的 | cān shù de |
| implicit/ɪmˈplɪsɪt/ | 隐式的 | yǐn shì de |
The derivative that never changes
- The function $e^x$ is unique: its derivative is itself. No other function has this property.
- This makes $e^x$ the foundation of growth models, radioactive decay, and compound interest.
从不改变的导数
- 函数 $e^x$ 是独一无二的:它的导数是它自己。没有别的函数有这个性质。
- 这使 $e^x$ 成为增长模型、放射性衰变和复利的基础。
Standard derivatives 标准导数
Worked example. $y = e^{3x}$. By the chain rule: $\dfrac{dy}{dx} = 3e^{3x}$.
标准导数
算例。 $y = e^{3x}$。由链式法则:$\dfrac{dy}{dx} = 3e^{3x}$。
The gradient at a point · 一个点处的斜率
gradient = dy/dx
Slide the point — the tangent · 相切 slope is the derivative, even for these new standard functions. · 滑动这个点——切线的斜率就是导数,即使对这些新的标准函数也是。
The derivative of ln x is 1/x. What is its value at x = 4? · ln x 的导数是 1/x。它在 x = 4 处的值是多少?
d/dx(ln x) = 1/x = 1/4 = 0.25. · d/dx(ln x) = 1/x = 1/4 = 0.25。
The derivative of eˣ is eˣ. What is its value at x = 0? · eˣ 的导数是 eˣ。它在 x = 0 处的值是多少?
d/dx(eˣ) = eˣ, and e⁰ = 1. · d/dx(eˣ) = eˣ,而 e⁰ = 1。
If y = e^(3x), then dy/dx = 3e^(3x). What is dy/dx at x = 0? · 如果 y = e^(3x),那么 dy/dx = 3e^(3x)。在 x = 0 处 dy/dx 是多少?
At x = 0: dy/dx = 3e⁰ = 3 × 1 = 3. · 在 x = 0:dy/dx = 3e⁰ = 3 × 1 = 3。
Product and quotient rules 商法则
- Product rule 乘积法则: $(uv)' = u'v + uv'$.
- Quotient rule: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$.
The product rule: when both $u$ and $v$ change, the total rate of change has two terms — one for each function changing while the other stays fixed.
Don't forget the product rule. The derivative of $uv$ is NOT $u'v'$. You must use $(uv)' = u'v + uv'$. This is one of the most common mistakes in differentiation.
乘积法则与商法则
- 乘积法则(product rule):$(uv)' = u'v + uv'$。
- 商法则(quotient rule):$\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$。
乘积法则:当 $u$ 和 $v$ 都改变时,总的变化率有两项——每项对应一个函数改变而另一个保持固定。
不要忘记乘积法则。 $uv$ 的导数不是 $u'v'$。你必须用 $(uv)' = u'v + uv'$。这是微分中最常见的错误之一。
The product rule says (uv)′ equals: · 乘积法则说 (uv)′ 等于:
The product rule: (uv)′ = u′v + uv′. · 乘积法则:(uv)′ = u′v + uv′。
The quotient rule for (u/v)′ is (u′v + uv′)/v². · (u/v)′ 的商法则是 (u′v + uv′)/v²。
The quotient rule is (u′v − uv′)/v² (minus, not plus in the numerator). · 商法则是 (u′v − uv′)/v²(分子中是减,不是加)。
Parametric 参数的 and implicit 隐式的 differentiation
- Parametric: $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$.
- Implicit: differentiate every term in $x$, chain-rule the $y$ terms, then solve for $\dfrac{dy}{dx}$.
参数微分与隐函数微分
- 参数(parametric):$\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$。
- 隐函数(implicit):对 $x$ 中的每一项微分,对 $y$ 项用链式法则,然后解出 $\dfrac{dy}{dx}$。
If x = t² and y = t³, then dy/dx = (dy/dt)/(dx/dt) = 3t²/2t = 3t/2. At t = 4, dy/dx = ? · 如果 x = t² 而 y = t³,那么 dy/dx = (dy/dt)/(dx/dt) = 3t²/2t = 3t/2。在 t = 4,dy/dx = ?
At t = 4: dy/dx = 3(4)/2 = 6. · 在 t = 4:dy/dx = 3(4)/2 = 6。
Worked example — product rule
- $y = x^2 \sin x$. Let $u = x^2$, $v = \sin x$.
- $u' = 2x$, $v' = \cos x$.
- $\dfrac{dy}{dx} = 2x\sin x + x^2\cos x$.
算例——乘积法则
- $y = x^2 \sin x$。设 $u = x^2$,$v = \sin x$。
- $u' = 2x$,$v' = \cos x$。
- $\dfrac{dy}{dx} = 2x\sin x + x^2\cos x$。
You've got it
- learn: $\dfrac{d}{dx}e^x = e^x$, $\dfrac{d}{dx}\ln x = \dfrac{1}{x}$, $\dfrac{d}{dx}\sin x = \cos x$, $\dfrac{d}{dx}\cos x = -\sin x$
- product rule $(uv)' = u'v + uv'$; quotient rule $\dfrac{u'v - uv'}{v^2}$
- parametric: $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$; implicit: chain-rule the $y$ terms
你掌握了
- 学会:$\dfrac{d}{dx}e^x = e^x$,$\dfrac{d}{dx}\ln x = \dfrac{1}{x}$,$\dfrac{d}{dx}\sin x = \cos x$,$\dfrac{d}{dx}\cos x = -\sin x$
- 乘积法则 $(uv)' = u'v + uv'$;商法则 $\dfrac{u'v - uv'}{v^2}$
- 参数:$\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$;隐函数:对 $y$ 项用链式法则