數學挑戰題 (5) -- 微分 (難) (40分)

Question

微分 :
(1)
y = 7^ x

(2)
y = 2^x (sin x)

(3)
y = x ^ (log x) (x > 0)

Answer 1

請看以下做法:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/Diffchallenge.jpg

Answer 2

(1)
y=7^x
ln y=x ln 7
d(ln y)/dx=d(x ln 7)/dx
(dy/dx)/y=ln 7
dy/dx=7^x ln 7

(2)
Let u=2^x
ln u=x ln 2
(du/dx)/u=ln 2
du/dx=2^x ln 2
dy/dx=(sin x) du/dx+u d(sin x)/dx
=sin x 2^x ln 2+2^x cos x
=ln 2 (sin x 2^x+cos x)

(3)
y=x^(log x)
ln y=log x ln x
d(ln y)/dx=d(log x ln x)/dx
(dy/dx)/y=ln x d(log x)/dx+log x d(ln x)/dx
(dy/dx)/y=ln x/(x ln 10)+log x/x
dy/dx=[x ln (x-10)+(log x)/x]x^(log x)

2007-02-21 17:23:54 補充：
Sorry第二題有錯誤sin x 2^x ln 2十2^x cos x=2^x (sin x ln 2十cos x)