科普视频
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例1
> [!question] 题干 > 对于常数a，有： > $ > \begin{cases} > > x=a(t-\sin{t}) \\ > y=a(1-\cos{t}) > > \end{cases} > $ > > 求y对x的导数
> [!done]- 解 > $ > \begin{aligned} > > &\begin{cases} > x=a(t-\sin{t}) \\ > y=a(1-\cos{t}) > \end{cases}\\ > > &\Downarrow{}&&两边微分\\ > > &\begin{cases} > dx=a\cdot d(t-\sin{t}) \\ > dy=a\cdot d(1-\cos{t}) > \end{cases}\\ > > &\Downarrow{}\\ > > &\begin{cases} > dx=a(dt-\cos{t}dt) &&根据d(\sin{x})=\cos{x} dx\\ > dy=a(0+\sin{t}dt) &&根据d(\cos{x})=-\sin{x}dx\\ > \end{cases}\\ > > &\Downarrow{}&&两个式子相除\\ > > \frac{dy}{dx}&=\frac{a\sin{t}dt}{a(1-\cos{t}dt)}\\ > > &=\frac{\sin{t}}{1-\cos{t}}&&约分\\ > &=\cot{\frac{t}{2}}&&可用万能公式或二倍角化简\\ > > \end{aligned} > $