The case of interest here is where X = ℝ^{n} and Y = ℝ^{m}, the function being defined on an open subset of
ℝ^{n}. Of course this all generalizes to arbitrary vector spaces and one considers the matrix taken
with respect to various bases. As above, f will be defined and differentiable on an open set
U ⊆ ℝ^{n}.

The matrix of Df

(x)

is the matrix having the i^{th} column equal to Df

(x )

e_{i} and so it is only necessary
to compute this. Let t be a small real number such that both

f (x + tei)− f (x) − Df (x)(tei) o(t)
------------t--------------= --t-

Therefore,

f (x-+-tei)−-f-(x)-= Df (x)(e )+ o(t)
t i t

The limit exists on the right and so it exists on the left also. Thus

∂f (x) f (x + te )− f (x)
----- ≡ lim -------i--------= Df (x)(ei)
∂xi t→0 t

and so the matrix of the derivative is just the matrix which has the i^{th} column equal to the i^{th} partial
derivative of f. Note that this shows that whenever f is differentiable, it follows that the partial derivatives
all exist. It does not go the other way however as discussed later.

Theorem 17.3.1Let f : U ⊆ F^{n}→ F^{m}and suppose f is differentiable at x. Then all the partialderivatives

∂fi(x)
∂xj

exist and ifJf

(x)

is the matrix of the linear transformation, Df

(x)

with respect to thestandard basis vectors, then the ij^{th}entry is given by

∂fi
∂xj

(x)

also denoted as f_{i,j}or f_{i,xj}. It is the matrixwhose i^{th}column is

∂f (x) f (x + te )− f (x)
----- ≡ lim -------i-------.
∂xi t→0 t

Of course there is a generalization of this idea called the directional derivative.

Definition 17.3.2In general, the symbol

D f (x)
v

is defined by

f (x+ tv)− f (x)
ltim→0 ------t--------

where t ∈ ℝ. In case

|v|

= 1 and the norm is the standard Euclidean norm, this is called the directionalderivative. More generally, with no restriction on the size of v and in any linear space, it is calledthe Gateaux derivative. f is said to be Gateaux differentiable at x if there exists D_{v}f

(x)

suchthat

f (x+ tv)− f (x)
lit→m0 ---------------= Dvf (x )
t

where v → D_{v}f

(x)

is linear. Thus we say it is Gateaux differentiable if the Gateaux derivative exists for each v andv → D_{v}f

What if all the partial derivatives of f exist? Does it follow that f is differentiable? Consider the
following function,f : ℝ^{2}→ ℝ,

{ -2xy2 if (x,y) ⁄= (0,0)
f (x,y) = x +y .
0 if (x,y) = (0,0)

Then from the definition of partial derivatives,

lim f (h,0)−-f (0,0)= lim 0−-0-= 0
h→0 h h→0 h

and

lim f (0,h)−-f (0,0)= lim 0−-0-= 0
h→0 h h→0 h

However f is not even continuous at

(0,0)

which may be seen by considering the behavior of the function
along the line y = x and along the line x = 0. By Lemma 17.1.4 this implies f is not differentiable.
Therefore, it is necessary to consider the correct definition of the derivative given above if you want to get a
notion which generalizes the concept of the derivative of a function of one variable in such a way as to
preserve continuity whenever the function is differentiable.