6.4. THE MATRIX OF THE DERIVATIVE 147

Theorem 6.4.1 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 6.4.2 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

Here is an interesting application which is used a lot in introductory courses on multi-variable calculus.

Theorem 6.4.3 Suppose U is an open set in a normed linear space X and f : U→Rhas a Gateaux derivative Dv f at x∈U and that for all x̂ sufficiently close to x, on the linethrough x having direction vector v it follows that f (x)≥ f (x̂)( f (x)≤ f (x̂)). In otherwords, f has a local maximum/minimum at x when restricted to the line t → x+ tv, thenDv f (x) = 0. If D f (x) exists and f has a local max/min at x for all v, then D f (x) = 0.

Proof: Consider h(t) = f (x+ tv) . Then from single variable calculus,

h′ (0) = Dv f (x) = 0.

In case f is differentiable, then for every v,

0 = Dv f (x) = limt→0

f (x+ tv)− f (x)t

= limt→0

D f (x)(tv)+o(t)t

= D f (x)v

Since this holds for every v it follows that D f (x) = 0. ■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)

.

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.