These notes introduce the notion of derivatives for functions of several variables.
It is worthwhile to rst recall the derivative of a real-valued function of a single
variable. Let f : R ! R, and xed x0 2 R. The classical de nition of f having a
derivative f0(x0) at x0 is that the limit
Noti...
These notes introduce the notion of derivatives for functions of several variables.
It is worthwhile to first recall the derivative of a real-valued function of a single
variable. Let f : R → R, and fixed x0 ∈ R. The classical definition of f having a
derivative f 0 (x0 ) at x0 is that the limit
0 f (x) − f (x0 )
f (x0 ) = lim
x→x0 x − x0
exists. Denoting the limit as a = f 0 (x0 ), we can rewrite this as
f (x) − f (x0 ) − a(x − x0 )
lim = 0.
x→x0 x − x0
Notice that any linear map from R to itself is just multiplication by some constant a.
This is the important conceptual leap we must make: the derivative of a function
at a point is not a number, or a vector. It is a linear transformation. The
derivative of a function f at the point x0 is the linear transformation which best
approximates f near x0 . Indeed, you’re already familiar with this concept from
Taylor’s theorem, but might not have thought about it using the language above.
Now we naturally arrive at the proper definition of a derivative:
Definition 1. Let f : Rn → Rm and x0 ∈ Rn . We say f has a derivative at x0 if
there is some linear map A : Rn → Rm such that
f (x) − f (x0 ) − A(x − x0 )
lim = 0.
x→x0 |x − x0 |
If this limit exists for the linear map A, we write Df |x0 = A as the (total) derivative
of f at the point x0 .
We can rewrite this expression as, near x0 ,
f (x) = f (x0 ) + A(x − x0 ) + r(x),
where
n m |r(x − x0 )|
A : R → R is linear, and lim = 0.
x→x0 |x − x0 |
1
, We call r(x − x0 ) the sublinear remainder of f at x0 . It will be convenient later to
define the auxiliary remainder function r̄ : [0, ∞) → [0, ∞) by
Notice that −r̄(|x − x0 |) ≤ r(x − x0 ) ≤ r̄(|x − x0 |), and we still have the estimate
r̄(|x − x0 |)
lim = 0.
x→x0 |x − x0 |
Also, not that r̄(ρ) is a monotone function of ρ, so that for ρ∗ > ρ > 0 we have
r̄(ρ∗ ) ≥ r̄(ρ).
We emphasize here that it is important to remember that the derivative of f at
the point x0 is not a number or a vector, it is the linear transformation which
best approximates f near x0 .
Proposition 1. If f is differentiable at x0 then it is also continuous.
Proof. Write x = x0 + v, and use the notation above. Then
We conclude limx→x0 (f (x)) = f (x0 ).
At this point we state some basic rules of differentiation.
Theorem 2. Let f and g be differentiable functions.
1. D(f + cg) = Df + cDg for all real numbers c.
2. D(g ◦ f ) = (Dg) ◦ (Df )
3. D(f · g) = Df · g + f · Dg (Notice the order of operations!)
The second two statements require some clarification. In the second statement,
we consider f : Rn → Rm and g : Rm → Rk . Then the composition maps Rn to Rk ,
and so does it’s derivative. The theorem states that the derivative of a composition
is the composition of derivatives. (This statement is certainly too elegant to not
be true!) In the third statement, the product in question is a bilinear operation we
will explain inside the proof.
2
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