求根号t/(t+1)的导数,用对数微分法,书中答案是：1/(2根号t)(t+1)^3/2

Question

求根号t/(t+1)的导数,用对数微分法,
书中答案是：1/(2根号t)(t+1)^3/2

Answer 1

两边取对数得
ln(y) = ln(t) / 2 - ln(t+1) /2
两边对t求导,由链式法则：
{ln(y)]' = 1/y * y' = 1/(2t) - 1/(2t+2) = 1 / [2(t+1)t]
所以y' = y/[2(t+1)t] = 1/(2*t^(1/2))(t+1)^3/2