Second Derivatives of Parametric Equations · 参数方程的二阶导数
The second derivative of a parametric curve
- We found $\tfrac{dy}{dx}$ for a parametric curve. Now: how curved is it? That needs $\tfrac{d^2y}{dx^2}$.
- The catch: you can't just divide the second derivatives in $t$.
- Instead, differentiate the first slope $\tfrac{dy}{dx}$ with respect to $t$, then divide by $\tfrac{dx}{dt}$ again.
- It's the parametric slope rule, applied a second time to $\tfrac{dy}{dx}$.
参数曲线的二阶导数
- 我们求了参数曲线的 $\tfrac{dy}{dx}$。现在:它弯曲多少?那需要 $\tfrac{d^2y}{dx^2}$。
- 陷阱:你不能只把 $t$ 的两个二阶导数相除。
- 而是把一阶斜率 $\tfrac{dy}{dx}$ 关于 $t$ 求导,再除以 $\tfrac{dx}{dt}$。
- 这是把参数斜率规则第二次应用到 $\tfrac{dy}{dx}$ 上。
The correct formula
- Treat the first derivative $\tfrac{dy}{dx}$ as a new function of $t$, and take its parametric slope:
-
$$\frac{d^2y}{dx^2}=\frac{\dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right)}{\dfrac{dx}{dt}}$$
- Differentiate $\tfrac{dy}{dx}$ with respect to $t$, then divide by $\tfrac{dx}{dt}$ (same denominator as before).
- The concavity's sign comes from this, just like an ordinary $f''$.
正确的公式
- 把一阶导数 $\tfrac{dy}{dx}$ 当作 $t$ 的新函数,取它的参数斜率:
-
$$\frac{d^2y}{dx^2}=\frac{\dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right)}{\dfrac{dx}{dt}}$$
- 把 $\tfrac{dy}{dx}$ 关于 $t$ 求导,再除以 $\tfrac{dx}{dt}$(与之前相同的分母)。
- 凹凸性的符号由此得出,就像普通的 $f''$。
Concavity of a parametric curve · 参数曲线的凹凸性
y = ax³ + bx
The parametric second derivative tells concavity — computed by differentiating $\tfrac{dy}{dx}$ in · 入 $t$, then dividing by $\tfrac{dx}{dt}$. · 参数二阶导数指示凹凸性 —— 通过对 $\tfrac{dy}{dx}$ 在 $t$ 中求导得到，然后除以 $\tfrac{dx}{dt}$。
The parametric second derivative $\dfrac{d^2y}{dx^2}$ equals... · 参数二阶导数 $\dfrac{d^2y}{dx^2}$ 等于...
Differentiate $\tfrac{dy}{dx}$ in · 入 $t$, then divide by $\tfrac{dx}{dt}$. · 对 $\tfrac{dy}{dx}$ 求导，然后除以 $t$，然后除以 $\tfrac{dx}{dt}$.
Order the steps to find the parametric second derivative. · 排序步骤以找到参数二阶导数。
Slope, differentiate in $t$, divide by $\tfrac{dx}{dt}$. · 斜率，在 $t$ 中对它求导，除以 $\tfrac{dx}{dt}$。
Why it isn't the obvious ratio
- It is not $\dfrac{d^2y/dt^2}{d^2x/dt^2}$ — that shortcut is wrong.
- $\tfrac{dy}{dx}$ is already a quotient of $t$-derivatives, so differentiating it needs the quotient rule (in $t$), not a naive second-derivative ratio.
- The right process always goes through $\tfrac{d}{dt}\big(\tfrac{dy}{dx}\big)$ first.
- Skipping that gives the classic wrong answer.
为何不是那个显然的比
- 它不是 $\dfrac{d^2y/dt^2}{d^2x/dt^2}$——那个捷径是错的。
- $\tfrac{dy}{dx}$ 本身已是 $t$ 导数的商,所以对它求导需要(关于 $t$ 的)商法则,而非简单的二阶导数之比。
- 正确过程总是先经过 $\tfrac{d}{dt}\big(\tfrac{dy}{dx}\big)$。
- 跳过它就得到经典的错误答案。
The parametric second derivative equals $\dfrac{d^2y/dt^2}{d^2x/dt^2}$. · 参数二阶导数等于 $\dfrac{d^2y/dt^2}{d^2x/dt^2}$。
That naive ratio is wrong. · 那个简单的比值是错误的。
The two-step routine
- 1. Find $\tfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$ and simplify.
- 2. Differentiate that with respect to $t$, then divide by $\tfrac{dx}{dt}$.
- Concave up where $\tfrac{d^2y}{dx^2}>0$, concave down where $<0$ — same as always.
- Keep the same denominator $\tfrac{dx}{dt}$ in both steps.
两步套路
- 1. 求 $\tfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$ 并化简。
- 2. 把它关于 $t$ 求导,再除以 $\tfrac{dx}{dt}$。
- $\tfrac{d^2y}{dx^2}>0$ 处上凹,$<0$ 处下凹——一如既往。
- 两步中保持同一分母 $\tfrac{dx}{dt}$。
For · 支持 $x=t^2$, $y=t^3$ ($\tfrac{dy}{dx}=\tfrac{3t}{2}$), $\dfrac{d^2y}{dx^2}=$ · 对于 $x=t^2$, $y=t^3$（$\tfrac{dy}{dx}=\tfrac{3t}{2}$）, $\dfrac{d^2y}{dx^2}=$
$\frac{d}{dt}(\tfrac{3t}{2})=\tfrac32$; divide by $2t$: $\tfrac{3}{4t}$. · $\frac{d}{dt}(\tfrac{3t}{2})=\tfrac32$；除以 $2t$：$\tfrac{3}{4t}$。
After differentiating $\tfrac{dy}{dx}$ in · 入 $t$, you divide by $\dfrac{dx}{dt}$, the ____ denominator as the first derivative. · 在对 $\tfrac{dy}{dx}$ 关于 $t$ 求导后，你除以 $\dfrac{dx}{dt}$，即作为一阶导数的 ____ 分母。
Both steps divide by $\tfrac{dx}{dt}$. · 两步都除以 $\tfrac{dx}{dt}$。
The sign of the parametric second derivative tells you the curve's... · 参数二阶导数的符号告诉你曲线的...
Positive → concave up, negative → concave down. · 正 → 凹向上，负 → 凹向下。
The second derivative is not $\dfrac{d^2y/dt^2}{d^2x/dt^2}$. You must differentiate the first derivative $\tfrac{dy}{dx}$ with respect to $t$ and then divide by $\tfrac{dx}{dt}$ again. Using the naive ratio of second derivatives is the single most common parametric mistake.
二阶导数不是 $\dfrac{d^2y/dt^2}{d^2x/dt^2}$。你必须把一阶导数 $\tfrac{dy}{dx}$ 关于 $t$ 求导,再除以 $\tfrac{dx}{dt}$。用简单的二阶导数之比是参数题最常见的单一错误。
For $x=t^2,\ y=t^3$ (so $\tfrac{dy}{dx}=\tfrac{3t}{2}$), find $\tfrac{d^2y}{dx^2}$.
- $\dfrac{d}{dt}\!\left(\dfrac{3t}{2}\right)=\dfrac{3}{2}$.
- Divide by $\dfrac{dx}{dt}=2t$: $\dfrac{d^2y}{dx^2}=\dfrac{3/2}{2t}=\dfrac{3}{4t}$.
- (Not $\tfrac{6t}{2}=3t$, which the wrong ratio would give.)
对 $x=t^2,\ y=t^3$(即 $\tfrac{dy}{dx}=\tfrac{3t}{2}$),求 $\tfrac{d^2y}{dx^2}$。
- $\dfrac{d}{dt}\!\left(\dfrac{3t}{2}\right)=\dfrac{3}{2}$。
- 除以 $\dfrac{dx}{dt}=2t$:$\dfrac{d^2y}{dx^2}=\dfrac{3/2}{2t}=\dfrac{3}{4t}$。
- (不是错误比会给出的 $\tfrac{6t}{2}=3t$。)
The parametric second derivative is $\dfrac{d^2y}{dx^2}=\dfrac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$ — differentiate the first slope $\tfrac{dy}{dx}$ in $t$, then divide by $\tfrac{dx}{dt}$ again. It is not $\frac{d^2y/dt^2}{d^2x/dt^2}$. Its sign gives concavity as usual.
参数二阶导数是 $\dfrac{d^2y}{dx^2}=\dfrac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$——把一阶斜率 $\tfrac{dy}{dx}$ 关于 $t$ 求导,再除以 $\tfrac{dx}{dt}$。它不是 $\frac{d^2y/dt^2}{d^2x/dt^2}$。其符号照常给出凹凸性。