函数y=13x^6+13x+arcsin9.x的导数计算步骤

1、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)。

2、一阶导数计算对y=13x^6+13x+arcsin9/x求一阶导数，有：dy/dx=13*6x^5+13+(9/x)'/√[1-(9/x)^2]=13*6x^5+13+(-9/x^2)/√[1-(9/x)^2]=78x^5+13-9/[x√(x^2-81)]。

3、二阶导数计算对dy/dx=78x^5+13-9/[x√(x^2-81)]继续对x求导有：dy^2/dx^2=78*5x^4+9*[√(x^2-81)+x*2x]/[x^2(x^2-81)]=390x^4+9*[√(x^2-81)+2x^2]/[x^2(x^2-81)]

4、三阶导数计算∵dy^2/dx=390x^4+9*[√(x^2-81)+2x^2]/[x^2(x^2-81)]，∴dy^3/dx^3=1560x^3+9*{[x/√(x^2-81)+4x][x^2(x^2-81)]-[√(x^2-81)+2x^2](4x^3-2*81x)}/[x^4(x^2-81)^2]=1560x^3+9*{[1/√(x^2-81)+4][x^2(x^2-81)]-2[√(x^2-81)+2x^2](2x^2-81)}/[x^3(x^2-81)^2]

5、=1560x^3+9*{[1陴鲰芹茯+4√(x^2-81)][x^2(x^2-81)]-2[(x^2-81)+2x^2*√(x^2-81)](2x^2-81)}/[x^3*√(x^2-81)^5]=1560x^3+9*[(x^2-81)(2*81-3x^2)-4x^2*√(x^2-81)]/[x^3*√(x^2-81)^5]=1560x^3+9*[(2*81-3x^2)*(x^2-81)-4x^4*√(x^2-81)]/[x^3*√(x^2-81)^5]=1560x^3+9*[(2*81-3x^2)*√(x^2-81)-4x^4]/[x^3*(x^2-81)^2]。