常用數學與微積分公式定理( 1 / 7 )常用數學公式常 用 數 學 公 式axdt txspecial casesbxyxyx yxyxrxxr)ln()ln( ),ln( ), ln( )ln()lnlnln( )lnlnln()ln= = +=+= z10100:aeeejbexceyyyeyyyjxyy).,cossin)exp( )exp(ln )lnln(exp( )ln1027182818281=+=sin()sincoscossinsin()sincoscossincos()coscossinsincos()coscossinsinsincossin()sin()cossinsin()sin()coscoscos()cos()sinsincos()cos()()/(ABABABABABABABABABABABABABABABABABABABABABABABABassumexAByABthenAxyBx+=+=+=+=+=+=+= +=+=RST=+=1 2 1 2 1 2 1 2 2 +RST+=+=+=+= +yxyxyxyxyxyxyxyxyxyxyxyxy)/sinsinsincossinsincossincoscoscoscoscoscossinsin2222222222222sinsincoscoscossincossincoscos,sincos222211212 212 2222222=+=常用數學與微積分公式定理( 2 / 7 )常用微分公式常 用 微 分 公 式d f ggdff dgdudud uudC dxdCC constantd xCdxdxd xC()()(:)()()=+=+=+=+121200de dxedee dxdx dxxududux xxx=lnln11dx dxnxdxnxdxdx dxxdxxdxd dxxxdxxdxn nnn= FHGIKJ=FHGIKJ=112 222221111d xyydxxdyd x ym xy dxn yx dym ydxn xdyd x y xydy xxdyydx xdx yydxxdy ymnmnnmmnmn()()=+=+ += FHGIKJ=FHGIKJ=ch111122dx dxxdxx dxdx dxxdxx dxdx dxxdxx dxdx dxxdxx dxdx dxxxdxxx dxdx dxxxdxxx dxsincossincoscossincossintansectanseccotcsccotcscsecsectansecsectancsccsc cotcsccsc cot= = = = = = 2222dx dxxdxxdxdx dxxdx xdxdx dxx xdxx xdxdx dxxdxxdxdx dxxdx xdxdx dxx xdxx xdxsinsintantansecseccoscoscotcotcsccsc = =+= = + +=122112211221122112211221111 1 11 111 111 1 11 111常用數學與微積分公式定理( 3 / 7 )微積分定理與公式微 積 分 定 理 與 公 式d dxf xg xd dxf xd dxg xd dxf ggd dxffd dxgf gfgf gf gfgf g gg xyy uuu xdy dxdy dudu dx( )( )( )( )( )( )( )=+=+ LNMOQP = =20chain rule : if andthen d F x dxf xf x dxF xCd dxf t dtf xad dxf x dxf xdf x dxf x dxf x dxdf x dxd f x dxdxf xCd f xf xCCyy xdyydxuu x yduu xdxu ydyu dvu vv duax( )( )( )( )( )( ),:( )( )( )( )( )( )( )( )( )( )(:)( )( , )=+=+=+= + RS| T|= zzzzzzzzzconstant.integral constant( integral by parts )( )( )d dxf x t dtf x b xdb dxf x p xdp dxxf x t dt p xb xp xb x( , ), ( ),( )( , )( )( )zz=+ bgbgdC dx dx dxnxn n=01de dxede dxaeda dxa ax xchain ruleax axx x= =ln常用數學與微積分公式定理( 4 / 7 )微積分定理與公式微 積 分 定 理 與 公 式dx dxxdx dxxdx dxxdx dxxdx dxxdx dxxdx dxxdx dxxdx dxxxdx dxxxdx dxxxdx dxxchain rulechain rulechain rulechain rulechain rulechain rulesincossincoscossincossintansectanseccotcsccotcscsecsectansecsectancsccsccotcsccsc= = = = = = = = = 2222cotxdx dxxdx dxxdx dxxdx dxxdx dxxdx dxxdx dxxdx dxxdx dxx xdx dxxchain rulechainrulechain rulechain rulechain rulesinsin()coscos()tantan ()cotcot ()secsec(= = = =+ =+= + = += =12121212121212121211111111 111 11111 xdx dxx xdx dxxxchain rule)csccsc()2121211111= =* Taylors series expansion :常用數學與微積分公式定理( 5 / 7 )微積分定理與公式微 積 分 定 理 與 公 式f xfa nxaf afaxafaxafaxaf(x)is aninfinitely differentiable functionsomeimportant expansions:ex nxxxxx nxxxxxn nnxnnnnn( )( ) !()( )( ) !()( ) !()( ) !().!sin() ()!cos( )( )=+= += +=+=+=023 302321035712311231 21357?= += +=1 21246111 212 32024623) ()!nnnpx nxxxxpxp pxp ppxbgbgbgbg?Binomial formulaxyC x yC xyLeibniz s formulad dxf gf gC fgwhere Cnkn knkn knkn kn k knkn n kknnn knkn kn kkn:! !( )( )()+=FHGIKJ=bgbg bgbg000x dxnxCndx xxCnn=+=+zz1 111()lne dxaeCa dxaaCaxaxxx=+=+zz11 ln常用數學與微積分公式定理( 6 / 7 )微積分定理與公式微 積 分 定 理 與 公 式1111 1 11 1 1111212121 212121=+=+=+ +=+=+=+zz zzzzxdxxC xdxxCxdxxCxdxxCx xdxxC x xdxxCsincostancotseccscebxdxe ababxbbxCebxdxe ababxbbxCaxaxaxaxcoscossinsinsincos=+ +=+ +z z2222sincoscossintanln secsectancotln sincsccotsecln sectansectancscln csccotcsccotsincoscossintanln seccotln sinxdxxCxdxxCxdxxCxdxxCxdxxCxdxxCxdxxxCxdxxCxdxxxCxdxxCxdxxCxdxxCxdxxCxdxzzzzzzzzzzzzzz= +=+=+=+=+= +=+=+= += += +=+=+=2222111111xCxdxxxCxdxxxC+=+= +zzsecln sectancscln csccot11