Question

高等数学，x^x+根号下(1+x^2)的高阶求导，求大神。。。。

Answer 1

追问

是求高阶导数，不是一次求导，一次求导俺也会

Answer 2

f(x)=x^x +根号(1+x^2)
g(x)=x^x h(x)=根号(1+x^2) f(x)=g(x)+h(x)
主要是对g(x) 求导是难点
lng(x)=xlnx
g'(x)/g(x)=lnx+x/x =lnx+1
g'(x)=g(x)[lnx+1]
h'(x)=1/2 *2x *1/根号(1+x^2) =x/根号(1+x^2)
所以f'(x)=g(x)[lnx+1]+x/根号(1+x^2) =x^x *[lnx+1] +x/根号(1+x^2]

追问

不是求一阶导，是求高阶导数，n阶导。。。一次导，我也会

追答

k(x)=x^x[lnx+1]=g(x)[lnx+1]
k'(x)=g'(x)[lnx+1] +g(x)/x =g(x)[lnx+1]^2 +g(x)/x =g(x)[[lnx+1]^2+1/x]
k''(x)=g'(x) [[lnx+1]^2+1/x] +g(x) * [2[lnx+1]/x -1/x^2]
=g(x)[lnx+1] [[lnx+1]^2+1/x]+g(x)*[2[lnx+1]/x -1/x^2]
=g(x)[[lnx+1]^3 +[lnx+1]/x+2[lnx+1]/x-1/x^2]
=g(x)[[lnx+1]^3 +3[lnx+1]/x -1/x^2]
k'''(x)=g'(x) [ [lnx+1]^3 +3[lnx+1]/x-1/x^2] +g(x)[3[lnx+1]^2 /x +3[1-lnx-1]/x^2+2/x^3]
=g(x) [[lnx+1]^4+3[lnx+1]^2 /x -[lnx+1]/x^2] +g(x)[3[lnx+1]^2 /x +3[1-lnx-1]/x^2+2/x^3]
=g(x)[ [nx+1]^4+6[lnx+1]^2/x -[4lnx+1]/x^2 +2/x^3]
规律不强哦，很难算

Answer 3

幂指函数取指数即可，=e^xln(x)这样你应该会了吧