For any function f(x,y), the first partial derivatives are represented by

f f— = fx and — = fyx 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 ysin 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

If each of the partial derivatives of f belongs to class C1, then we say thatf belongs to class C2.

x sin y > 0.

We can calculate higher order (and mixed) partial derivatives:

f — ( — ) = — ( fx ) = ( fx )x = fxx =x x x

f — ( — ) = — ( fy ) = ( fy )y = fyy =y y y

f — ( — ) = — ( fx ) = ( fx )y = fxy =y x y

f — ( — ) = — ( fy ) = ( fy )x = fyx =x y x

1– — x2

1– —— = – csc2 y sin2 y

0

0

f 1 ff(x,y) = log(x sin y), — = fx = — and — = fy = cot y

x x y

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)

f(x , y+y ) – f(x , y)—————————

y

f(x+x , y) – f(x , y)—————————

x

fx(x0 , y0+y) – fx(x0 , y0)——————————

y

(x0+x , y0+y)

(x0+x , y0)

(x0 , y0+y)

(x0 , y0)

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

fx(x , y)

fy(x , y)

Consider

fxy(x0 , y0)

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

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).

f(x0+x , y0+y) – f(x0+x, y0)substitute ————————————— in place of fy(x0+x , y0) .

y