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【正文】 ??? xxx c o ss i n)(t a n ? x2cosxx c o s)(s in ? )( c o ss in ?? xx ? x2cosx2cos x2sin? x2sec?xcos?xx c o tc s c??類似可證 : ,c s c)(c o t 2 xx ??? .ta ns e c)( s e c xxx ??機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 返回 結(jié)束 Derivatives of sec(x), csc(x) and cot(x) ? All are found by applying the product and/or quotient rules and using known derivatives of sin(x) and cos(x). xxfxxxfxxxf2cs c)( co t39。t ans ec)( s ec39。 For a posite function, its derivative is found by taking the derivative of the outer function, with respect to the inner function, times the derivative of the inner function with respect to x. ? If the position consists of 3 or more functions, continue to take the derivative of the next inner function, with respect to the function within it, until, finally, the derivative is taken with respect to x. 在點(diǎn) x 可導(dǎo) , ??????? ?? lim0x xyxyx ?????? 0l i mdd復(fù)合函數(shù)求導(dǎo)法則 定理 3. 在點(diǎn) 可導(dǎo) 復(fù)合函數(shù) 且 )()(dd xgufxy ???在點(diǎn) x 可導(dǎo) , 證 : )( ufy ?? 在點(diǎn) u 可導(dǎo) , 故 )(lim0ufuyu??????uuufy ??????? ?)(（當(dāng) 時(shí) ) 故有 )()( xguf ???uy?? ???? )( uf)0()( ??????????? xxuxuufxy ?機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 返回 結(jié)束 求下列函數(shù)的導(dǎo)數(shù) 212s inxxy??222222222212c o s)1(22)1()20(2)1(212c o s1212c o sxxxxxxxxxxxxxxy????????????????????????例如 , ?xydd)()()( xvuf ?? ??????yuvx?uydd ?vudd xvdd關(guān)鍵 : 搞清復(fù)合函數(shù)結(jié)構(gòu) , 由外向內(nèi)逐層求導(dǎo) . 推廣 ： 此法則可推廣到多個(gè)中間變量的情形 . 機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 返回 結(jié)束 例 . 求 解 : ,1111???????xxxxy .y?2122 2 ??? xxy? 12 ??? x1??? y1212 ??x)2( x?112 ???xx例 . 設(shè) ),0( ???? aaaxy xaa axa解 : 1??? aaa xay aa ax ln? 1?? axaaa xa ln?求 .y?aa x ln?機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 返回 結(jié)束 例 . 求 解 : ,1a r c t a n 2s i n 2 ?? xey x .y?1a r c t a n) ( 2 ??? xy) (2s i n xe?2s in xe 2cos x? x2?21x 1212 ??xx2?x2? 1a r c t a n 2 ?x2s in xe2cos x2s in xe112 ??xx關(guān)鍵 : 搞清復(fù)合函數(shù)結(jié)構(gòu) 由外向內(nèi)逐層求導(dǎo) 機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 返回 結(jié)束 例 . 設(shè) 求 ,1111ln411a r c t a n21222???????xxxy .y?解 : ??y22 )1(1121x?? 21 xx??)11l n ()11l n ( 22 ????? xx?111412 ???x 21 xx??1112 ???x?21 xx???2121xx?? 221x??21x?23 1)2(1xxx ????機(jī)動(dòng) 目錄 上頁(yè) 下頁(yè) 返回 結(jié)束 Find the derivative (note this is the position of 3 functions, therefore there will be 3 “pieces” to the chain.) )co tcs c3(])cs c(5[])cs cco s [ (39。 Approximations ? dx is the differential of x, graphically it is the change in the x of the tangent to the curve (dy/dx) ? dy is the differential of y, graphically is corresponds to the change in the y of the tangent to the curve (dy/dx) 微分在近似計(jì)算中的應(yīng)用 )()( 0 xoxxfy ??????當(dāng) x? 很小時(shí) , )()( 00 xfxxfy ????? xxf ??? )( 0xxfxfxxf ?????? )()()( 000xxx ??? 0令使用原
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