中四微积分若ln(x+y)=x,求dy/dx.(答案以x表示)若y=e^(x+y),求dy/dx.(答案以y表示)

Question

中四微积分
若ln(x+y)=x,求dy/dx.(答案以x表示)
若y=e^(x+y),求dy/dx.(答案以y表示)

Answer 1

1、ln(x+y) = x
这既是复合函数(Composite Function),又是隐函数(implicit Function)
必须用链式求导法(Chain Rule)
(1+dy/dx)/(x+y)=1,ie.dy/dx = x+y-1
2、y = e^(x+y)
本题类似于第一题,一样用Chain Rule
dy/dx = [e^(x+y)](1+dy/dx)
= y(1+dy/dx)
dy/dx - ydy/dx = y
dy/dx = y/(1-y)