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.

f 1 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

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+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

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).