# Basic properties of derivatives and differentials

*Properties of derivatives and differentials.*

Derivative of a composite function.

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
*
x
*
_{
0
}
, 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
*
x
*
_{
0
}
, and a
function
*
g
*
has a derivative at a point
*
f
*
(
*
x
*
_{
0
}
*
*
), then a composite function
*
h
*
has also a derivative at a point
*
x
*
_{
0
}
*
*
, calculated by the formula:

*h’*(

*x*

_{ 0 }

*) =*

*g’*(

*f*(

*x*

_{ 0 }

*) ) ·*

*f’*(

*x*

_{ 0 }

*) .*