Question

高数求隐函数的偏导数，第二大题的1,2,3,4,题

Answer 1

cos(x^2-y)是复合函数求导
设x^2-y=u
根据复合函数求导法则
(cosu)'=-sinu*u'
所以cos(x^2-y)求导变成[－sin(x^2-y)](x^2-y)'
x*(根号x^2-a^2）/2+ln3
求导=根号(x^2-a^2)/2+x*[(x^2-a^2)^(1/2)]'/2
=根号(x^2-a^2)/2+x*(x^2-a^2)'*(x^2-a^2)^(1/2-1)/2*2
=根号(x^2-a^2)/2+x^2/2倍根号(x^2-a^2)
通分之后
=(x^2-a^2+x^2)/2倍根号(x^2-a^2)
=(2x^2-a^2)/2倍根号(x^2-a^2)
因为根号(x^2-a^2)=(x^2-a^2)^(1/2)
所以x*(根号x^2-a^2）/2+ln3=x*(x^2-a^2)^(1/2)/2+ln3
根据复合函数求导的运算法则,
上式求导
=(x)'*(x^2-a^2)^(1/2)/2+x*[(x^2-a^2)^(1/2)]'/2
=(x^2-a^2)^(1/2)/2+x*(1/2)*(x^2-a^2)^(1/2-1)*(x^2-a^2)'/2
=(x^2-a^2)^(1/2)/2+x*(1/2)*(x^2-a^2)^(1/2-1)*2x/2
=(x^2-a^2)^(1/2)/2+x^2*(x^2-a^2)^(-1/2)/2