本文主要通过函数和求导规则，介绍函数y=9x^8+6x+arcsin4/x的一阶、二阶和三阶导数计算步骤。本题应用到的函数导数有y=x^a，dy/dx=ax^a-1；y=bx，dy/dx=b；y=arcsincx，dy/dx=c/√(1-c^2*x^2)。

对y=9x^8+6x+arcsin4/x求一阶导数，有：

dy/dx=9*8x^7+6+(4/x)'/√[1-(4/x)^2]

=9*8x^7+6+(-4/x^2)/√[1-(4/x)^2]

=63x^7+6-4/[x√(x^2-16)]。

对dy/dx=63x^7+6-4/[x√(x^2-16)]

继续对x求导有：

dy^2/dx^2

=63*7x^6+4*[√(x^2-16)+x*2x]/[x^2(x^2-16)]

=441x^6+4*[√(x^2-16)+2x^2]/[x^2(x^2-16)]

∵dy^2/dx=441x^6+4*[√(x^2-16)+2x^2]/[x^2(x^2-16)]，

∴dy^3/dx^3

=2646x^5+4*{[x/√(x^2-16)+4x][x^2(x^2-16)]-[√(x^2-16)+2x^2](4x^3-2*16x)}/[x^4(x^2-16)^2]

=2646x^5+4*{[1/√(x^2-16)+4][x^2(x^2-16)]-2[√(x^2-16)+2x^2](2x^2-16)}/[x^3(x^2-16)^2]

=2646x^5+4*{[1+4√(x^2-16)][x^2(x^2-16)]-2[(x^2-16)+2x^2*√(x^2-16)](2x^2-16)}/[x^3*√(x^2-16)^5]

=2646x^5+4*[(x^2-16)(2*16-3x^2)-4x^2*√(x^2-16)]/[x^3*√(x^2-16)^5]

=2646x^5+4*[(2*16-3x^2)*(x^2-16)-4x^4*√(x^2-16)]/[x^3*√(x^2-16)^5]

=2646x^5+4*[(2*16-3x^2)*√(x^2-16)-4x^4]/[x^3*(x^2-16)^2]。