1. Partial derivative
z = f ( x, y )
ϕ ( t ) = f ( a, t )
tan α =
α
a
b
dϕ ( t )
dt
|t =b

2. Let f ( x1 , x2 ,K , xn ) be defined on M ⊆ En and let ( a1 , a2 ,K , an ) ∈ M .
Define ϕ i ( t ) = f ( a1 , a2 ,K , ai −1 , t , ai +1 ,K xn )
If ϕ i ( t ) has a derivative at t = ai , we call its value
the partial derivative of f ( x1 , x2 ,K , xn ) by xi at ( a1 , a2 ,K , an ) .

3. We use several different ways to denote the partial derivative
of f ( x1 , x2 ,K , xn ) by xi at ( a1 , a2 ,K , an ) :
∂f ( x1 , x2 ,K , xn )
∂xi
|[ x1 , x2 ,K, xn ] =[ a1 ,a2 ,K,an ]
∂f ( a1 , a2 ,K , an )
∂xi
f x'i ( x1 , x2 ,K , xn ) |[ x1 , x2 ,K, xn ] =[ a1 ,a2 ,K, an ]
f x'i ( a1 , a2 ,K , an )
Note that the symbol ∂
reads "d" and is not
identical to the Greeek
delta δ
!!!

4. f x'i ( x1 , x2 ,K , xn ) in n variables that assigns to each
The function
( a1 , a2 ,K , an )
f ( x1 , x2
at which the partial,K , xn )
derivative of
exists the value of such a partial derivative is then called the
f ( x1 , x ,K , xn )
partial derivative of 2
by xi
Sometimes this function is also denoted
∂f ( x1 , x2 ,K , xn )
∂xi

5. When calculating such a partrial derivative, we use the
following practical approach:
When calculating the partial derivative of f ( x1 , x2 ,K , xn ) by
xi, we think of every variable other than xi as of a constant
parameter and treat it as such. Then we actually calculate the
"ordinary" derivative of a function in one variable.

6. Calculate all the partial derivatives of the following functions
f ( x, y, z ) = 3 x 3 yz 4 − 7 xz + 12 y 5 z 2
f ( x, y, z ) = ln x 2 + y 2 + z 2
x
arctan  
 y
f ( x, y ) =
x2 + y 2

7. Partial derivatives of higher orders
If a partial derivative is viewed as a function it may again be
differentiated by the same or by a different variable to become a
partial derivative of a higher order. Theoretically, there may be a
partial derivative of an arbirary order if it exists.
f x''i x j ( x1 , x2 ,K , xn ) , f x''i xi ( x1 , x2 ,K , xn ) , etc.
Notation:
∂ 2 f ( x1 , x2 ,K , xn ) ∂ 2 f ( x1 , x2 ,K , xn )
,
, etc.
2
∂xi ∂x j
∂xi

8. ∂ f ( x1 , x2 ,K , xn )
2
∂xi ∂x j
∂ f ( x1 , x2 ,K , xn )
2
=
?
∂x j ∂xi

9. Schwartz' theorem
Let X = [ x0 , y0 ] be an internal point of the domain of f ( x, y )
''
and let, in a neighbourhood N ( X , δ ) of X, f yx ( x, y ) exist
''
and be continuus at X = [ x0 , y0 ] . Then f xy ( x, y ) exists and
''
''
f xy ( x0 , y0 ) = f yx ( x0 , y0 )
Similar assertions also hold for functions in more than two
variables and for higher order derivatives.

10. Schwartz' theorem
Let X = [ x0 , y0 ] be an internal point of the domain of f ( x, y )
''
and let, in a neighbourhood N ( X , δ ) of X, f yx ( x, y ) exist
''
and be continuus at X = [ x0 , y0 ] . Then f xy ( x, y ) exists and
''
''
f xy ( x0 , y0 ) = f yx ( x0 , y0 )
Similar assertions also hold for functions in more than two
variables and for higher order derivatives.