题目

高数求导（dy/dx）习题
设由下列方程确定y是x的函数,求dy/dx
(1)cos(x^2 +y)=x
求下列参数方程所确定的函数y=f(x)的导数dy/dx
(1)x=(e^t)sint,y=(e^t)cost.
(1)-[1+2xsin(x^2 +y)]/[sin(x^2 +y)]
(2)cost-sint/sint+cost

提问时间：2020-11-28

答案

(1)cos(x^2 +y)=x
-sin(x^2 + y)[2x + dy/dx]=1
dy/dx = -2x - csc(x^2 + y)
= -[1+2xsin(x^2 +y)]/[sin(x^2 +y)]
(2)x=(e^t)sint,y=(e^t)cost.
dy = [(e^t)cost - (e^t)sint]dt
dx = [(e^t)sint + (e^t)cost]dt
两式相除,得：
dy/dx = [(e^t)cost - (e^t)sint]/[(e^t)sint + (e^t)cost]
= [cost - sint]/[sint + cost]
或继续消去参数：
= [1 - tant]/[tant + 1]
= [1 - x/y]/[x/y + 1]
= [y - x]/[x + y]