7.4. THE MATRIX OF THE DERIVATIVE 189

Theorem 7.4.1 Let f : U ⊆ Fn→ Fm and suppose f is differentiable at x. Then allthe partial derivatives ∂ fi(x)

∂x jexist and if Jf (x) is the matrix of the linear transformation,

Df (x) with respect to the standard basis vectors, then the i jth entry is given by ∂ fi∂x j

(x)

also denoted as fi, j or fi,x j . It is the matrix whose ith column is

∂f (x)

∂xi≡ lim

t→0

f (x+ tei)−f (x)t

.

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

Definition 7.4.2 In general, the symbol Dvf (x) is defined by

limt→0

f (x+ tv)−f (x)t

where t ∈ F. In case |v|= 1,F = R, and the norm is the standard Euclidean norm, this iscalled the directional derivative. More generally, with no restriction on the size of v and inany linear space, it is called the Gateaux derivative. f is said to be Gateaux differentiableat x if there exists Dvf (x) such that

limt→0

f (x+ tv)−f (x)t

= Dvf (x)

where v → Dvf (x) is linear. Thus we say it is Gateaux differentiable if the Gateauxderivative exists for each v and v→ Dvf (x) is linear. Note that ∂f(x)

∂xi= Deif (x).

1

What if all the partial derivatives of f exist? Does it follow that f is differentiable?Consider the following function, f : R2→ R,

f (x,y) ={ xy

x2+y2 if (x,y) ̸= (0,0)0 if (x,y) = (0,0)

.

Then from the definition of partial derivatives,

limh→0

f (h,0)− f (0,0)h

= limh→0

0−0h

= 0

and

limh→0

f (0,h)− f (0,0)h

= limh→0

0−0h

= 0

However f is not even continuous at (0,0) which may be seen by considering the behaviorof the function along the line y = x and along the line x = 0. By Lemma 7.2.3 this impliesf is not differentiable. Therefore, it is necessary to consider the correct definition of thederivative given above if you want to get a notion which generalizes the concept of thederivative of a function of one variable in such a way as to preserve continuity wheneverthe function is differentiable.

What if the one dimensional derivative in the definition of the Gateaux derivative existsfor all nonzero v? Is the function differentiable then? Maybe not. See Problem 12 in theexercises for example.

1René Gateaux was one of the many young French men killed in world war I. This derivative is named afterhim, but it developed naturally from ideas used in the calculus of variations which were due to Euler and Lagrangeback in the 1700’s.