题目

求导数,
(10) y=ln√x+√lnx (12)y=sinnx (14)y=sin^nx (16)y=cos^3 x/2
(18) y=lntanx/2 (20) y=lnlnx (22)y=1/(cos^n x) (24)y=sec^2 x/a+csc^2 x/a

提问时间：2021-03-21

答案

(10) y=ln√x+√lnx=ln(x^0.5)+(lnx)^0.5=0.5*ln x+(lnx)^0.5
y'=0.5/x+0.5*(lnx)^(-0.5)/x=(0.5/x)*(1+1/((lnx)^0.5)
(12)y=sinnx
y'=(cosnx)*n=n*(cosnx)
(14)y=sin^nx=(sinx)^n
y'=n*(sinx)^(n-1)*cosx=n*cosx*(sinx)^(n-1)
(16)y=cos^3 x/2=(cos x/2)^3
y'=3*(cos x/2)^2*(sin x/2)/2=(3/2)*(sin x/2)*(cos x/2)^2
(18) y=lntanx/2=ln(tan(x/2))
y'=1/(tanx/2)*(sec(x/2))^2/2=(sec(x/2))^2/(2*tanx/2)=sin(x/2)/(2*(cos(x/2))^3)
(20) y=lnlnx=ln(lnx)
y'=1/(lnx)/x=1/(x*lnx)
(22)y=1/(cos^n x)=1/(cos x)^n=(cos x)^(-n)
y'=-n*(cos x)^(-n-1)*(-sin x)=(n*sin x)/((cos x)^(n+1))
(24)y=sec^2 x/a+csc^2 x/a=(sec x/a)^2+(csc x/a)^2
y'=2*(sec x/a)*(tan x/a)*(sec x/a)/a=(2/a)*(tan x/a)*(sec x/a)^2=(2*sin x/a)/((cos x/a)^3)