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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