4.4. THE USUAL FORM OF THE CHAIN RULE 95

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 4.1.4 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.

4.4 The Usual Form of the Chain RuleLet z≡ g(y) and let y = f(x). Assuming Dg(f(x)) exists and Df(x) both exist and then wehave x ∈U ⊆Rn and y ∈V ⊆Rm where U,V are open sets and f(V )⊆V with g : V →Rp.What is the matrix of g◦ f(x) = g(y)? Say g has values in Rp. From the chain rule above,and the description of the matrix of the derivative in Theorem 4.3.1,(

∂z∂x1

∂z∂x2

· · · ∂z∂xn

)= (4.5)

(∂z∂y1

∂z∂y2

· · · ∂z∂ym

)(∂y∂x1

∂y∂x2

· · · ∂y∂xn

)Now from the way we multiply matrices, to find the i jth entry of the matrix on the right,one multiplies the ith row of the left matrix with the jth column of the matrix on the right.Thus the i jth entry of the matrix on the right is

∑k

∂ zi

∂yk

∂yk

∂x j

and by the chain rule, Theorem 4.2.1, this equals the i jth entry of the matrix of 4.5. That is,

∂ zi

∂x j= ∑

k

∂ zi

∂yk

∂yk

∂x j

This is stated as the following proposition.

Proposition 4.4.1 Let z≡ g◦ f(x) and let y≡ f(x) , then assuming Df(x) exists andDg(f(x)) exists, then ∂ zi

∂x j= ∑k

∂ zi(y)∂yk

∂yk∂x j

assuming that all functions make sense. That isf : U→ f(U)⊆V and g : V →Rp for U,V open sets in Rn and Rm respectively. Also, sincethis holds for each i, ∂z

∂x j= ∑k

∂z∂yk

∂yk∂x j

.

Some people like to dispense with the summation sign and write instead ∂z∂x j

= ∂z∂yk

∂yk∂x j

where it is understood that summation takes place on the repeated index.