設 f(x) 為可導函數。已知 f(8) = 6 及 f'(

設 f(x) 為一可導函數。已知 f(8)=6及 f'(8)=－2
(a) 設 u(x) = x^3。求 du/dx
du/dx=dx^3/dx=3x^2
(b) 求當 x= 8 時 d/dx[f(x)]^3 的值。
d[f(x)]^3/dx
=｛d[f(x)]^3/df(x)｝*df(x)/dx
=3[f(x)]^2*f'(x)
d[f(x)]^3/dx｜x=8
=3*6^2*(－2)
=－216
(c) 若 g(x) = f(x^3) 及 h(x) =[f(x^3)]^3，利用 (a)及(b)的結果，求
(i) g'(2)
dg(x)/dx
=df(x^3)/dx
=[df(x^3)/dx^3]*dx^3/dx
=3x^2f'(x^3)
g'(2)=3*4*(－2)=－24
(ii) h'(2)
dh(x)/dx
=d[f(x^3)]^3/dx
=｛d[f(x^3)]^3/df(x^3)｝*｛df(x^3)/dx^3｝*(dx^3/dx)
=｛3[f(x^3)]^2｝*f'(x^3)*(3x^2)
h'(2)=3*36*(－2)*12=－2592