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)