let
be an open subset of
, and let
be a function. then
is differentiable at
with derivative
if the limit
another derivative notation
to denote the
th derivative of a function we write
, this reads "the
th derivative of
with respect to
"
an alternative definition of the derivative in one dimension
let
be an open subset of
and
a function. then
is differentiable at
, with derivative
, if and only if
the letter
denotes the "change in";
is the change in the function;
is the change in the variable
. the function
that multiplies
by the derivative
is thus a linear function of the change in
.
let
be an open subset and let
be a mapping; let
be a point in
. if there exists a linear transformation
such that
then
is differentiable at
, and
is unique and is the derivative of
at
, denoted
, and whose transformation matrix is of dimensions
.
if
is differentiable at
, then all partial derivatives of
exist, and the matrix representing
is
(the jacobian matrix).
since the square of an
matrix is another
matrix, and such a matrix can be "identified" with
, this could be written as a function
. this is one time when a linear transformation is easier to deal with than the corresponding matrix. we denote by
the set of
matrices, and consider the squaring map
in this case we can compute the derivative without computing the jacobian matrix. we shall see that
is differentiable and that its derivative
is the linear transformation that maps
to
:
the first thing to realize is that the map
is a linear transformation. the asseration is that
since
,
this gives
so
is indeed the derivative.
vector calculus derivative table
let
be open.
- if
is a constant function, then
is differentiable, and its derivative is
(the zero linear transformation
, represented by the
matrix filled with 0's)
- if
is linear, then it is differentiable everywhere, and its derivative at all points
is
, i.e.,
.
if are scalar-valued functions differentiable at , then the vector-valued mapping is differentiable at , with derivative
conversely, if is differentiable at , each is differentiable at , and .
if are differentiable at , then so is , and
if and are differentiable at , then so is , and the derivative is given by
if and are differentiable at , and , then so is , and the derivative is given by
if are both differentiable at , then so is the dot product , and (as in one dimension)