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运算法则

Kamimika...大约 1 分钟学习笔记

运算法则

加减乘除

[u(x)±v(x)]=u(x)±v(x) [u(x) \pm v(x)]' = u'(x) \pm v(x)

[u(x)v(x)]=u(x)v(x)+u(x)v(x) [u(x) v(x)]' = u'(x)v(x) + u(x)v'(x)

[u(x)v(x)]=u(x)v(x)u(x)v(x)v2(x) [\dfrac{u(x)}{v(x)}]' = \dfrac{u'(x)v(x) - u(x)v'(x)}{v^2(x)}

提示

提示: 乘法和除法中都有 u(x)v(x)u'(x)v(x)u(x)v(x)u(x)v'(x)

复合(链式法则)

[f(g(x))]=f(g(x))g(x)=f(x)x=g(x)g(x) \begin{align} [f(g(x))]' &= f'(g(x)) \cdot g'(x) \\ &= f'(x)|_{x=g(x)} \cdot g'(x) \end{align}

dydx=dydududx \dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx}

亦见链式法则#一阶导

幂指函数(对数求导法)

乘除法 \to 加减法 (简化计算) 多个乘除法/幂指函数 \to 对数求导法

y=f(x)g(x)    lny=g(x)lnf(x)同时求导,    yy=g(x)lnf(x)+g(x)f(x)f(x)    y=[g(x)lnf(x)+g(x)f(x)f(x)]f(x)g(x) y = f(x)^{g(x)} \\ \implies \ln y = g(x) \ln f(x) \\ 同时求导, \implies \dfrac{y'}{y} = g'(x) \ln f(x) + g(x) \dfrac{f'(x)}{f(x)} \\ \implies y' = [g'(x) \ln f(x) + g(x) \dfrac{f'(x)}{f(x)}] \cdot f(x)^{g(x)}

反函数

(f1)(x)=1f(y) (f^{-1})'(x) = \dfrac{1}{f'(y)}

dydx=1dxdy \dfrac{dy}{dx} = \dfrac{1}{\dfrac{dx}{dy}}

例如: y=arcsinxy = \arcsin xx=sinyx = \sin y 的反函数, dxdy=d(siny)dy=cosy\dfrac{dx}{dy} = \dfrac{d(\sin y)}{dy} = \cos y, dydx=1cosy=11sin2y=11x2\dfrac{dy}{dx} = \dfrac{1}{\cos y} = \dfrac{1}{\sqrt{1 - \sin^2 y}} = \dfrac{1}{\sqrt{1-x^2}} 注意最后要尽量写成以 xx 为自变量的形式

参数方程

{x=f(t),y=g(t)(tI) \begin{cases} x &= f(t), \\ y &= g(t) \end{cases} (t \in I)

dydx=f(t)g(t)=dydtdxdt \dfrac{dy}{dx} = \dfrac{f'(t)}{g'(t)} = \dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}

极坐标方程

{x=r(θ)cosθ,y=r(θ)sinθ(αθβ) \begin{cases} x &= r(\theta) \cos \theta, \\ y &= r(\theta) \sin \theta \end{cases} (\alpha \leq \theta \leq \beta)

dydx=y(θ)x(θ)=r(θ)sinθ+r(θ)cosθr(θ)cosθr(θ)sinθ \dfrac{dy}{dx} = \dfrac{y'(\theta)}{x'(\theta)} = \dfrac{r'(\theta) \sin \theta + r(\theta) \cos \theta}{r'(\theta) \cos \theta - r(\theta) \sin \theta}

注意

drdθ\dfrac{dr}{d\theta} 非切线斜率

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