BOA Systems : Information base

Differentials

The following general rules can be applied to all differentials, where f() and g() are functions.

^d/_dx( n f(x) )	=	n ^d/_dx( f(x) )
^d/_dx( f(x) + g(x) )	=	^d/_dx( f(x) ) + ^d/_dx( g(x) )
^d/_dx( f(x) g(x) )	=	f(x) ^d/_dx( g(x) ) + g(x) ^d/_dx( f(x) )
^d/_dx( f(x) / g(x) )	=	( g(x) ^d/_dx( f(x) ) - f(x) ^d/_dx( g(x) ) ) / g²(x)
^d/_dx( f( g(x) )	=	^d/_dg( f(x) ) ^d/_dx( g(x) )

Differentials of Common Functions

Function F(x) = y	Differential F'(x) = ^dy / _dx
xⁿ	n x^n-1
1/x	-1/x²
e^ax	a e^ax
Ln x	1/x
Log_ax	1/x Log_ae
sin ax	a cos ax
cos ax	-a sin ax
tan ax	a sec² ax
cosec x	- cot x cosec x
sec x	tan x sec x
cot x	- cosec² x
arcsin(x/a)	1 / Ö ( a² - x² )
arccos(x/a)	-1 / Ö( a² - x² )
arctan(x/a)	a / ( a² + x² )

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