V2EX implicit differentiation

Implicit Differentiation

定义 Definition

隐式求导:一种求导方法,适用于函数关系以隐式方程形式给出(如 \(F(x,y)=0\)),通过对等式两边同时对 \(x\) 求导,并把 \(y\) 视为 \(x\) 的函数(因此会出现 \(\mathrm{d}y/\mathrm{d}x\)),从而求出导数。

发音 Pronunciation (IPA)

/mplst dfrnien/

例句 Examples

We used implicit differentiation to find dy/dx for the circle equation.
我们用隐式求导来为圆的方程求 \(\mathrm{d}y/\mathrm{d}x\)。

Because the relationship between x and y was given implicitly, implicit differentiation was the quickest way to compute the slope without solving for y first.
由于 \(x\) 和 \(y\) 的关系是以隐式形式给出的,隐式求导是不先解出 \(y\) 就能计算斜率的最快方法。

词源 Etymology

implicit 源自拉丁语 implicitus,有“卷在一起的、含在内的”之意,在数学语境中引申为“未直接写成 \(y=f(x)\) 的形式”。differentiation 来自 differentia(差异),在微积分中指“求导/微分”。合起来表示“对隐式关系进行求导”。

相关词 Related Words

文学与著作中的用例 Literary Works

  • Calculus: Early Transcendentals(James Stewart)
  • Thomas’ Calculus(George B. Thomas 等)
  • Calculus(Michael Spivak)
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