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Some Important Derivative
Merging man and maths
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d d n c=0 where c is constant x = nxn -1 · dx dx ì d d d 2 x x x x · = · = · s i n c o s t a n s e c csc x= - csc xcot x ïï dx dx dx í d d ï · d cos x = - sin x · cot x = - csc2 x · sec x= sec xtan x ïî dx dx dx ì d 1 d 1 -1 d 1 -1 S i n x · = · Sec -1 x = ï T a n x · = dx 1 - x2 x x2 -1 ï dx dx 1 + x2 í d d -1 -1 -1 ï· d Cos -1 x = -1 C s c x · Cot -1x = · = ï dx dx dx 1+ x2 1- x2 x x2 - 1 î d 1 ì d x x · log a x = ïï· dx a = a ln a dx x ln a í d 1 ï· d e x = e x x = · ln ïî dx dx x ì d d d · tanh x = sech 2 x · sech x= - sech xtanh x ïï· dx sinh x = cosh x dx dx í d d ï· d cosh x = sinh x · coth x = - csch 2 x · csch x= - csch xcoth x ïî dx dx dx ì d 1 d -1 -1 d 1 · Sech -1 x = ï· dx Sinh x = 2 Tanh -1 x = · Tan dx x +1 x 1 - x 2 ï dx 1 - x2 í d d -1 1 -1 ï· d Cosh -1 x = 1 C s c h x · Coth -1 x = · = ï dx dx dx 1- x2 x 2 -1 x 1 + x 2 î
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Some Standard nth Derivative n
m -n
d m! ax + b) m = a n ( ax + b ) · n ( dx ( m - n )!
(-1)n -1 ( n - 1)! a n d n éln(ax + b) ùû = · n ë dx (ax + b)n d n p ö æ n · a x b a a x b n + = + + s i n ( ) s i n . ç dx n 2 ÷ø è
if m ³ n
n
( -1)
n! a n d æ 1 ö · = dx n çè ax + b ÷ø (ax + b)n +1 n
d n a x e = a n ea x · n dx d n p ö æ n · a x b a a x b n + = + + c o s ( ) c o s . ç dx n 2 ÷ø è
n æ d n ax 2 2 2 ax -1 b ö · e b x c a b e b x c n + = + + + s i n ( ) ( ) s i n t a n ç ÷ dx n aø è n æ d n ax 2 2 2 ax -1 b ö · e b x c a b e b x c n + = + + + c o s ( ) ( ) c o s t a n ç ÷ dx n aø è
Leibniz’s Theorem d n (n) ( n -1) ( n -2) n n n n n -1 n n ¢ ¢ ¢ ¢ u v C u v C u v C u v C u v C u v = + + + × × × × × × × × × × × × × × + + ( ) 0 1 2 1 n n dx n Made by Atiq ur Rehman Rehman (
[email protected] [email protected] ) http://www.mathcity.org Submit errors at http://www.mathcity.org/error