1. For any function f(x,y), the first partial derivatives are represented by
f f
— = fx and — = fy
x y
For example, if f(x,y) = log(x sin y), the first partial derivatives are
f f
— = fx = and — = fy =
x y 1 — x
cos y
—— = cot y
sin y
If a function f from Rn to R1 has continuous partial derivatives, we say that f belongs to class C1. We can see that f(x,y) = log(x sin y) belongs to class C1 when its domain is defined so that
x sin y > 0.
If each of the partial derivatives of f belongs to class C1, then we say that
f belongs to class C2.

2. f f
f(x,y) = log(x sin y), — = fx = — and — = fy = cot y
x x y
We can calculate higher order (and mixed) partial derivatives:
 f 
— ( — ) = — ( fx ) = ( fx )x = fxx =
x x x 1
– — x2
 f 
— ( — ) = — ( fy ) = ( fy )y = fyy =
y y y 1
– —— = – csc2 y
sin2 y
 f 
— ( — ) = — ( fx ) = ( fx )y = fxy =
y x y
 f 
— ( — ) = — ( fy ) = ( fy )x = fyx =
x y x

3. Let f(x,y) = sin(xy)
fx = fy =
fxx = fyy =
fxy =
fyx =
y cos(xy)
x cos(xy)
– y2 sin(xy)
– x2 sin(xy)
cos(xy) – xy sin(xy)
cos(xy) – xy sin(xy)

4. f(x+x , y) – f(x , y)
————————— x
(x0 , y0+y)
(x0+x , y0+y)
fx(x , y) 
f(x , y+y ) – f(x , y)
————————— y
fy(x , y) 
Consider
fxy(x0 , y0) 
(x0+x , y0)
(x0 , y0)
fx(x0 , y0+y) – fx(x0 , y0)
—————————— y
f(x0+x , y0) – f(x0 , y0)
Substitute ————————— in place of fx(x0 , y0) , and x
f(x0+x , y0+y) – f(x0 , y0+y)
substitute ————————————— in place of fx(x0 , y0+y) . x

5. Now consider
fy(x0+x , y0) – fy(x0 , y0)
fyx(x0 , y0)  —————————— x
f(x0 , y0+y) – f(x0 , y0)
Substitute ————————— in place of fy(x0 , y0) , and y
f(x0+x , y0+y) – f(x0+x, y0)
substitute ————————————— in place of fy(x0+x , y0) . y
Note that the results are the same in both cases suggesting that
fxy = fyx .
Look at Theorem 1 on page 183 (and note how this result can be extended to partial derivatives of any order).