同济高数 第2章 第2-4-*10题

教材习题

📝 题目

＊10．求下列参数方程所确定的函数的三阶导数 $\displaystyle \frac{\mathrm{d}^{3} y}{\mathrm{~d} x^{3}}$ ： （1）$\left\{\begin{array}{l}x=1-t^{2}, \\ y=t-t^{3} ;\end{array}\right.$ （2）$\left\{\begin{array}{l}x=\ln \left(1+t^{2}\right), \\ y=t-\arctan t .\end{array}\right.$

💡 答案解析

[AI解答]

[AI解答]

**（1）** 已知参数方程 $$ \begin{cases} x = 1 - t^2, \\ y = t - t^3. \end{cases} $$

先求一阶导数： $$ \frac{dy}{dx} = \frac{\displaystyle\frac{dy}{dt}}{\displaystyle\frac{dx}{dt}} = \frac{1 - 3t^2}{-2t} = \frac{3t^2 - 1}{2t}. $$

再求二阶导数： $$ \frac{d^2y}{dx^2} = \frac{\displaystyle\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\displaystyle\frac{dx}{dt}} = \frac{\displaystyle\frac{d}{dt}\left( \frac{3t^2 - 1}{2t} \right)}{-2t}. $$ 先计算分子： $$ \frac{d}{dt}\left( \frac{3t^2 - 1}{2t} \right) = \frac{(6t)(2t) - (3t^2 - 1)(2)}{4t^2} = \frac{12t^2 - 6t^2 + 2}{4t^2} = \frac{6t^2 + 2}{4t^2} = \frac{3t^2 + 1}{2t^2}. $$ 因此 $$ \frac{d^2y}{dx^2} = \frac{\displaystyle\frac{3t^2 + 1}{2t^2}}{-2t} = -\frac{3t^2 + 1}{4t^3}. $$

最后求三阶导数： $$ \frac{d^3y}{dx^3} = \frac{\displaystyle\frac{d}{dt}\left( \frac{d^2y}{dx^2} \right)}{\displaystyle\frac{dx}{dt}} = \frac{\displaystyle\frac{d}{dt}\left( -\frac{3t^2 + 1}{4t^3} \right)}{-2t}. $$ 先计算分子导数： $$ \frac{d}{dt}\left( -\frac{3t^2 + 1}{4t^3} \right) = -\frac{1}{4} \cdot \frac{(6t)(t^3) - (3t^2+1)(3t^2)}{t^6} = -\frac{1}{4} \cdot \frac{6t^4 - 9t^4 - 3t^2}{t^6} = -\frac{1}{4} \cdot \frac{-3t^4 - 3t^2}{t^6} = -\frac{1}{4} \cdot \frac{-3t^2(t^2+1)}{t^6} = \frac{3(t^2+1)}{4t^4}. $$ 因此 $$ \frac{d^3y}{dx^3} = \frac{\displaystyle\frac{3(t^2+1)}{4t^4}}{-2t} = -\frac{3(t^2+1)}{8t^5}. $$

所以（1）的结果为： $$ \boxed{-\frac{3(t^2+1)}{8t^5}}. $$

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**（2）** 已知参数方程 $$ \begin{cases} x = \ln(1+t^2), \\ y = t - \arctan t. \end{cases} $$

先求一阶导数： $$ \frac{dx}{dt} = \frac{2t}{1+t^2}, \quad \frac{dy}{dt} = 1 - \frac{1}{1+t^2} = \frac{t^2}{1+t^2}. $$ 因此 $$ \frac{dy}{dx} = \frac{\displaystyle\frac{t^2}{1+t^2}}{\displaystyle\frac{2t}{1+t^2}} = \frac{t}{2}. $$

再求二阶导数： $$ \frac{d^2y}{dx^2} = \frac{\displaystyle\frac{d}{dt}\left( \frac{t}{2} \right)}{\displaystyle\frac{dx}{dt}} = \frac{\displaystyle\frac{1}{2}}{\displaystyle\frac{2t}{1+t^2}} = \frac{1}{2} \cdot \frac{1+t^2}{2t} = \frac{1+t^2}{4t}. $$

最后求三阶导数： $$ \frac{d^3y}{dx^3} = \frac{\displaystyle\frac{d}{dt}\left( \frac{1+t^2}{4t} \right)}{\displaystyle\frac{dx}{dt}}. $$ 先计算分子： $$ \frac{d}{dt}\left( \frac{1+t^2}{4t} \right) = \frac{1}{4} \cdot \frac{(2t)(t) - (1+t^2)(1)}{t^2} = \frac{1}{4} \cdot \frac{2t^2 - 1 - t^2}{t^2} = \frac{1}{4} \cdot \frac{t^2 - 1}{t^2}. $$ 因此 $$ \frac{d^3y}{dx^3} = \frac{\displaystyle\frac{t^2 - 1}{4t^2}}{\displaystyle\frac{2t}{1+t^2}} = \frac{t^2 - 1}{4t^2} \cdot \frac{1+t^2}{2t} = \frac{(t^2 - 1)(1+t^2)}{8t^3} = \frac{t^4 - 1}{8t^3}. $$

所以（2）的结果为： $$ \boxed{\frac{t^4 - 1}{8t^3}}. $$

难度：★★☆☆☆

📋 详细解题步骤

步骤 1/3
目标：求一阶导数 dy/dx
对于参数方程，一阶导数公式为 dy/dx = (dy/dt)/(dx/dt)。计算 dx/dt = -2t，dy/dt = 1-3t^2，所以 dy/dx = (1-3t^2)/(-2t) = (3t^2-1)/(2t)。
公式：dy/dx = (dy/dt)/(dx/dt)
提示：注意符号处理，分子分母同时乘以-1可简化表达式。
步骤 2/3
目标：求二阶导数 d²y/dx²
二阶导数公式为 d²y/dx² = d(dy/dx)/dt / (dx/dt)。先求 d(dy/dx)/dt = d/dt[(3t^2-1)/(2t)] = (6t*2t - (3t^2-1)*2)/(4t^2) = (12t^2-6t^2+2)/(4t^2) = (6t^2+2)/(4t^2) = (3t^2+1)/(2t^2)。再除以 dx/dt = -2t，得 d²y/dx² = (3t^2+1)/(2t^2) / (-2t) = -(3t^2+1)/(4t^3)。
公式：d²y/dx² = d(dy/dx)/dt / (dx/dt)
提示：求导时使用商的导数公式，注意化简。
步骤 3/3
目标：求三阶导数 d³y/dx³
三阶导数公式为 d³y/dx³ = d(d²y/dx²)/dt / (dx/dt)。先求 d(d²y/dx²)/dt = d/dt[-(3t^2+1)/(4t^3)] = -1/4 * d/dt[(3t^2+1)/t^3] = -1/4 * [(6t*t^3 - (3t^2+1)*3t^2)/t^6] = -1/4 * [(6t^4 - 9t^4 - 3t^2)/t^6] = -1/4 * [(-3t^4-3t^2)/t^6] = -1/4 * [-3t^2(t^2+1)/t^6] = 3(t^2+1)/(4t^4)。再除以 dx/dt = -2t，得 d³y/dx³ = [3(t^2+1)/(4t^4)] / (-2t) = -3(t^2+1)/(8t^5)。
公式：d³y/dx³ = d(d²y/dx²)/dt / (dx/dt)
提示：注意符号和化简，最终结果可写成 -3(1+t^2)/(8t^5)。

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