微分 :
(1)
y = 7^ x
(2)
y = 2^x (sin x)
(3)
y = x ^ (log x) (x > 0)
數學挑戰題 (5) -- 微分 (難) (40分)
2007-02-21 8:10 pm
回答 (2)
2007-02-21 8:27 pm
參考: My Maths knowledge
2007-02-21 8:22 pm
(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)
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)
收錄日期: 2021-04-21 12:10:51
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