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 xn-1
1/x -1/x2
eax a eax
Ln x 1/x
Logax 1/x Logae
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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