✔ 最佳答案
用到公式
chain rule : y =f (u) = f (g (x) ) = > dy/dx = (dy/du) × (du/dx)
函數相乘微分公式 : d (f × g)/dx = (df/dx) × g+ f × (dg/dx)
1.
F = (sin^ 2 x) × (cos^2 x) / tan^2 x
=(sin^ 2 x) × (cos^2 x) / (sinx/cosx)^2
={(sin^ 2 x) × (cos^2 x) / {(sin^2 x)/(cos^2 x)}
=(cos^2 x) × (cos^2 x)
=cos^4 x
dF/dx = d cos^4 x /dx = { d cos^4 x /d cosx } × (d cosx /dx)
=4 cos^3 x × (-sinx)
=-4 sinx cos^3 x
2.
F = 1/cosx =(cosx) ^(-1)
dF/dx =d (cosx) ^(-1) /dx ={d (cosx) ^(-1) /d cosx} × (d cosx /dx)
= - (cosx) ^(-2) × (-sinx)
= sinx (cosx)^-2
=sinx /(1－ sin^2 x }
3.
F = sin^2 x * cos^2 x
dF/dx = d (sin^2 x * cos^2 x)/dx
= (d sin^2 x/dx) × (cos^2 x ) + (d cos^2 x/dx) × (sin^2 x )
={ (d sin^2 x/d sinx) × (d sinx/dx) }× (cos^2 x )
+{ (d cos^2 x/d cosx) × (d cosx /dx) }× (sin^2 x )
=2sinx cosx cos^2 x + 2cosx (-sinx) (sin^2 x)
=2 sinx cos^3 x － 2cosx sin^3 x
=2 sinx cosx (cos^2 x －sin^2 x)
2014-10-19 17:17:55 補充：
第三題是用到乘法公式跟chain rule嗎 ?
Ans : 是的，先乘法公式，再chain rule
2014-10-19 17:35:32 補充：
secx = 1/cosx
三角函數的六角形圖 ，應該有教過吧
http://sylworld.forum888.net/t310-topic
2014-10-19 18:22:20 補充：
1.
sinx與cosx微分推導
https://www.flickr.com/photos/25446201@N08/15385869490/
2.
lim (h→0) sinh/h =1 & lim (h→0) (cosh-1)/h =0 證明
http://webcai.math.fcu.edu.tw/calculus/calculus_html/3-5/triderivate.htm
2014-10-19 18:22:47 補充：
3.
常用三角函數公式.
http://wywu.pixnet.net/blog/post/27411230-%E5%B8%B8%E8%A6%8B%E4%B8%89%E8%A7%92%E5%
2014-10-19 18:30:15 補充：
3. 上面網址， 無法開啟，改用這個
常用三角函數公式
http://zh.wikibooks.org/zh-hant/%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B8
2014-10-20 11:54:33 補充：
dF/dx ={ (1)' cosx -1(cosx)'} /cos^2 x =sinx /cos^2 x
