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