这道题怎么求导的,r到底是常数还是变量,又为什么要把t换成r

Question

这道题怎么求导的,r到底是常数还是变量,又为什么要把t换成r

Answer 1

注意是对t求导,r是常数
F(t)=2∫(0到t)f(r²)r²dr / ∫(0到t)f(r²)rdr
对t求导
F'(t)={ [2∫(0到t)f(r²)r²dr]' *[∫(0到t)f(r²)rdr] -[2∫(0到t)f(r²)r²dr] * [∫(0到t)f(r²)rdr]' } / [∫(0到t)f(r²)rdr]²
显然[2∫(0到t)f(r²)r²dr]'= 2f(t²)t²,[∫(0到t)f(r²)rdr]'=f(t²)t
故
F'(t)= ｛2f(t²)t² *[∫(0到t)f(r²)rdr] -[2∫(0到t)f(r²)r²dr] * f(t²)t ｝/ [∫(0到t)f(r²)rdr]²
= 2t *f(t²) * [∫(0到t)f(r²)r(t-r)dr] / [∫(0到t)f(r²)rdr]²
就是你要的答案