本文主要通過函數和求導規則，介紹函數y=9x^8 6x arcsin4/x的一階、二階和三階導數計算步驟。本題應用到的函數導數有y=x^a，dy/dx=ax^a-1；y=bx，dy/dx=b；y=arcsincx，dy/dx=c/√(1-c^2*x^2)。
對y=9x^8 6x arcsin4/x求一階導數，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)]。
對dy/dx=63x^7 6-4/[x√(x^2-16)]
繼續對x求導有：
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]。
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