22.7. HIGHER ORDER DERIVATIVES 721

22.7 Higher Order DerivativesIf f : U ⊆ X → Y for U an open set, then

x→ Df(x)

is a mapping from U to L (X ,Y ), a normed vector space. Therefore, it makes perfect senseto ask whether this function is also differentiable.

Definition 22.7.1 The following is the definition of the second derivative.

D2f(x)≡ D(Df(x)) .

Thus,Df(x+v)−Df(x) = D2f(x)v+o(v) .

This impliesD2f(x) ∈L (X ,L (X ,Y )) , D2f(x)(u)(v) ∈ Y,

and the map(u,v)→ D2f(x)(u)(v)

is a bilinear map having values in Y . In other words, the two functions,

u→ D2f(x)(u)(v) , v→ D2f(x)(u)(v)

are both linear.The same pattern applies to taking higher order derivatives. Thus,

D3f(x)≡ D(D2f(x)

)and D3f(x) may be considered as a trilinear map having values in Y . In general Dkf(x)may be considered a k linear map. This means the function

(u1, · · · ,uk)→ Dkf(x)(u1) · · ·(uk)

has the propertyu j→ Dkf(x)(u1) · · ·(u j) · · ·(uk)

is linear.Also, instead of writing

D2f(x)(u)(v) , or D3f(x)(u)(v)(w)

the following notation is often used.

D2f(x)(u,v) or D3f(x)(u,v,w)

with similar conventions for higher derivatives than 3. Another convention which is oftenused is the notation

Dkf(x)vk