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2.4.2 由参数方程所确定的函数的求导法
设变量x,y之间的函数关系由方程组
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00066003.jpg?sign=1734395368-Esp3v3M8CXJY9VPtDfUwMDR2ffEk02Av-0-ee283ebd9ccc4b89ada46f606939c6d4)
所确定,则称此函数关系所表示的函数为由参数方程所确定的函数.此时 (已知x=φ(t),y=ψ(t)都可导,φ'(t)≠0)的求法为:
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00066004.jpg?sign=1734395368-menJGR1qm1UJ9Lhia0kctxWSyQbOm58n-0-e81185bfe54ca5c941de6a920c7bc3c9)
例5 设参数方程.求
.
解 由参数方程的求导公式,得
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00066007.jpg?sign=1734395368-itZ95qpxRlmfHyoxcvyvkDHTxtCPuzRX-0-6bc0150d662e65b2359437fc6c8e0ff2)
例6 已知摆线的参数方程为.
(1)求在任何点处的切线的斜率;
(2)求在处的切线方程.
解 (1)摆线在任何点处的切线的斜率为
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00066010.jpg?sign=1734395368-3aOwTfoMfded3SkdmYpRKcxqDR1HFXZk-0-5a8bc4709b394272b8dd346ae2e67fdc)
(2)当时,摆线上对应的点为
,在此点的切线斜率为
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00066013.jpg?sign=1734395368-nharNI1lHyloQQA1IAsbdS0zyreMovBO-0-c3fece12f7c348c05840a610e5c61c30)
于是,切线方程为
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00066014.jpg?sign=1734395368-sMb4gfzxtAB9OOwwBINakvgVbPbqSiUk-0-2acce46cac2bfe087e594613e3bb17e0)
即
![](https://epubservercos.yuewen.com/C83605/14615860104561706/epubprivate/OEBPS/Images/img00067001.jpg?sign=1734395368-B2XgMnHNphcLpwNJbEPWJA9IMdjTcK33-0-8ed743a4771ecde7109bd3fcde19343d)