94 CHAPTER 4. THE DERIVATIVE

4.3 The Matrix of the DerivativeThe case of interest here is where X = Rn and Y = Rm, the function being defined onan open subset of Rn. Of course this all generalizes to arbitrary vector spaces and oneconsiders the matrix taken with respect to various bases. As above, f will be defined anddifferentiable on an open set U ⊆ Rn.

The matrix of Df(x) is the matrix having the ith column equal to Df(x)ei and so it isonly necessary to compute this. Recall that for Jf(x) this matrix,

Jf(x)v = Df(x)v

for any v where on the left the meaning is matrix multiplication. Jf(x) is an m×n matrixand it is multiplying v, a vector of Rn on the left. This is the matrix taken with respect tothe standard basis vectors. Let t be a small real number. Then

f(x+ tei)− f(x)−Df(x)(tei)

t=

o(t)t

Therefore,f(x+ tei)− f(x)

t= Df(x)(ei)+

o(t)t

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

∂ f(x)∂xi

≡ limt→0

f(x+ tei)− f(x)t

= Df(x)(ei)

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

Theorem 4.3.1 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

≡ limt→0

f(x+ tei)− f(x)t

.

In particular, this says the same as saying that the i jth entry of this matrix is ∂ fi(x)∂x j

.

I will generally not distinguish between the linear transformation Df(x) and its matrixwith respect to the standard basis vectors Jf(x) when the setting is Rn and Rm.

If you take another partial derivative, it can be written as fxix j ≡ ∂

∂x j

∂ f∂xi

. This might bewritten as f,i j also. I assume the reader has seen partial derivatives in calculus.

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)

.