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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 );

(   v )  =  u  v  ,      d (   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 ) = gf ( x0 ) )   f ( x0 ) .

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