Basic properties of derivatives and differentials
Properties of derivatives and differentials.
Derivative of a composite function.
If u ( x )
≡ const
, then
u’ ( x ) ≡ 0 , du
≡ 0.
If u ( x ) and v ( x ) are differentiable
functions at a point x0
, then:
( c u )’ = c u’ ,
d ( c u ) = c du , ( c – const );
( u ±
v )’ = u’ ±
v’ , d ( u ±
v ) = du ± dv
;
( u v )’ = u’ v + u v’ , d (
u v ) = v du + u dv ;
Derivative of a composite function. Consider a composite function, argument of which is also a function: h ( x ) = g ( f (
x ) ). If a function f has a derivative at a point x0 , and a
function g has a derivative at a point f ( x0
), then a composite function h has also a derivative at a point x0
, calculated by the formula:
h’ ( x0
) = g’ ( f ( x0
) ) · f’ ( x0
) .
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