Chapter 22

Derivative of a Functions of Many Vari-ables

Linear functions were just discussed. The derivative of a nonlinear function of many vari-ables is a linear approximation to the function which is valid locally. You have f : U →Rm

where U is an open subset of Rn and the derivative at some point is T ∈ L (Rn,Rm) suchthat near the point x, f (x+v) is close to T (v)+f (x). This is the main idea.

22.1 The Derivative of Functions of One VariableFirst consider the notion of the derivative of a function of one variable.

Observation 22.1.1 Suppose a function f of one variable has a derivative at x. Then

limh→0

| f (x+h)− f (x)− f ′ (x)h||h|

= 0.

This observation follows from the definition of the derivative of a function of one variable,namely

f ′ (x)≡ limh→0

f (x+h)− f (x)h

.

Thus

limh→0

| f (x+h)− f (x)− f ′ (x)h||h|

= limh→0

∣∣∣∣ f (x+h)− f (x)h

− f ′ (x)∣∣∣∣= 0

Definition 22.1.2 A vector valued function of a vector v is called o(v) (referredto as “little o of v”) if

lim|v|→0

o(v)

|v|= 0. (22.1)

Thus for a function of one variable, the function f (x+h)− f (x)− f ′ (x)h is o(h).When we say a function is o(h), it is used like an adjective. It is like saying the function iswhite or black or green or fat or thin. The term is used very imprecisely. Thus in general,

o(v) = o(v)+o(v) , o(v) = 45×o(v) , o(v) = o(v)−o(v) ,etc.

When you add two functions with the property of the above definition, you get another onehaving that same property. When you multiply by 45, the property is also retained, as itis when you subtract two such functions. How could something so sloppy be useful? Thenotation is useful precisely because it prevents you from obsessing over things which arenot relevant and should be ignored.

Theorem 22.1.3 Let f : (a,b)→R be a function of one variable. Then f ′ (x) existsif and only if there exists p such that

f (x+h)− f (x) = ph+o(h) (22.2)

In this case, p = f ′ (x).

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