本文研究平面、空间直线在参数方程形式下，切线斜率（即导数）如何表示的问题。
$$\begin{gathered} 如上图所示。 \\ 设y=f(x)，x=\phi (t)，y=\psi (t) \\ 当t=t_0时，x=x_0，y=y_0，即点A坐标为(x_0,y_0) \\ 点A处的导数f^{\prime }(x_0)=\underset{\Delta x\to 0}{\lim }\frac{\Delta y}{\Delta x}=\underset{\Delta t\to 0}{\lim }\frac{\psi (t_0+\Delta t)-\psi (t_0)}{\phi (t_0+\Delta t)-\phi (t_0)} \\ =\underset{\Delta t\to 0}{\lim }\frac{\psi (t_0+\Delta t)-\psi (t_0)}{\Delta t}/\frac{\phi (t_0+\Delta t)-\phi (t_0)}{\Delta t} \\ =\underset{\Delta t\to 0}{\lim }\frac{\psi (t_0+\Delta t)-\psi (t_0)}{\Delta t}/\underset{\Delta t\to 0}{\lim }\frac{\phi (t_0+\Delta t)-\phi (t_0)}{\Delta t} \\ =\frac{{\psi }^{\prime }(t_0)}{{\phi }^{\prime }(t_0)} \\ 因此点A处的切线向量可表示为({\psi }^{\prime }(t_0),{\phi }^{\prime }(t_0)) \\ 而切线方程为y-y_0=\frac{{\psi }^{\prime }(t_0)}{{\phi }^{\prime }(t_0)}(x-x_0)，即\frac{y-y_0}{{\psi }^{\prime }(t_0)}=\frac{x-x_0}{{\phi }^{\prime }(t_0)} \\ 同理可得空间直线的点向式方程为： \\ \frac{y-y_0}{{\psi }^{\prime }(t_0)}=\frac{x-x_0}{{\phi }^{\prime }(t_0)}=\frac{z-z_0}{{\omega }^{\prime }(t_0)} \end{gathered}$$