n rr Properties if lim xa fxl ID: 253240
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www.mathportal.org \n \r\r Properties if lim() xa fxl ® = and lim()xa gxm ® = , then [ ] lim()()xa fxgxlm ® ±=± [ ] lim()()xa fxgxlm ® ×=× ()lim()xa fxl gxm ® = where 0 m ¹ lim()xa cfxcl ® ×=× 11 lim()xa fxl ® = where 0 l ¹ Formulas lim1 e ®¥ += () lim1 ne ®¥ += sin lim1 xx x ® = tan lim1 xx x ® = cos1 lim0 xx x ® - = 1 limnn n xaxa na xa - ®-=- 01 limln nxa a x ®-= \n \r Basic Properties and Formulas () () cfcfx ¢ ¢ = () ()() fgfxgx ¢ ¢¢ ±=+ Product rule () fgfgfg ¢ ¢¢ ×=×+× Quotient rule 2 ffgfg gg¢ ¢¢ ×-× Power rule ( ) 1 nn d xnx dx - = Chain rule ()()()() ()() d fgxfgxgx dx ¢¢ = Common Derivatives () 0 dc dx = () 1 dx dx = () sincos d xx dx = () cossin d xx dx =- () 2 21 tansec cos d xx dx x == () secsectan d xxx dx = () csccsccot d xx dx =- () 2 21 cotcsc sin d xx dx x =-=- ( ) 1 2 sindx x -=- ( ) 1 2 cosdx x =- ( ) 1 2 1 tan 1 dx x -= + ( ) ln xx aaa dx = ( ) xx d ee dx = () 1 ln,0 xx dxx =� () 1 ln,0 xx dxx =¹ () 1 log,0 ln xx dxxa =� www.mathportal.org \r Definitions and properties Second derivative 2 2 ddydy dxdx dx ¢¢=-Higher-Order derivative ()()()nnff ¢ () ( ) ()() nn fgfg +=+() ( ) ()() nn fgfg -=-Leibnizs Formulas () 2. fgfgfgfg ¢¢ ¢¢¢¢¢¢ ×=×+×+()33 fgfgfgfgfg ¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢¢ ×=×+×+×+× ()()()() ( ) ()()12... 12 n nnnn nn fgfgnfgfgfg --¢¢×=++++Important Formulas ()()() !!n mmn m xx mn - =- ( ) ( ) ! nn xn = ()() ( ) ( ) 1 11! log ln nnannx xa - -- = × ()() ( ) ( ) 1 11! ln x - -- () ( ) ln xxn aaa = ( ) ( ) n xx ee = ( ) ( ) ln mxnmxn amaa ()() sinsin 2 n n xx p =+ ()() coscos 2 n n xx p =+