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Partial Derivatives

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Consider a two variable function f(x,y), suppose we let only xvary while keeping y  fixed, say y = b, where b  is a constant. Then we are really considering a function of a single variable x, namely g(x) = f(x,b). If ghas a derivative at a, then we call it the partial derivative of fwith respect to xat  ( a,b) and denote it by  

Diff(f(a,b),x)(or f[x](a,b) ).

 Thus by the definition of a derivative we have:

diff(f(a,b),x) = limit((f(a+h,b)-f(a,b))/h,h = 0)

Similarly, the partial derivative of f with respect to y at ( a,b), Diff(f(a,b),y)(or f[y](a,b) ) is given by:

diff(f(a,b),y) = limit((f(a,b+k)-f(a,b))/k,k = 0)

Let's consider the function f(x,y) = 4-x^2-2*y^2 , the graphs below show the geometric meaning of Diff(f(a,b),x)and diff(f(a,b),y)  :

 [Maple Plot]
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[Maple Plot]
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Here is an alternate geometric interpretation for the partial derivative in terms of vector functions:

Let g(x) = f(x,b)and the graph Cof the function g(viewed on the plane y = b.) is the curve traced out by the vector function G (x)=( x, b, f(x,b)) whose vector derivative G '( x)=( 1, 0, diff(f(a,b),x)) is determined by diff(f(a,b),x). (That is, its "slope" is diff(f(a,b),x).)

[Maple Plot]
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similarly, the graph of h(x) = f(a,y)is the curve C[2]traced out by the vector function H (y)=( a, y, f(a,y)) whose vector derivative H '( b)=( 0, 1, diff(f(a,b),y)) is determined by diff(f(a,b),y).

[Maple Plot]
3DView

Now let's take a look at a function   f which is not continuous at ( 0,0), but  diff(f(x,y),x)exists at ( 0, 0). Let f(x,y) = x*y/(x^2+y^2)for all ( x,y) except at ( 0,0), and f(0,0) = 0.

[Maple Plot]
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[Maple Plot]
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As we can see from the graphs above, f(x,y) = 0  along y = 0, x = 0and f(x,y) = 1for all (x, y) except at (0, 0) along y = x. So that f(x,y)is not continuous at (0,0) (limit(f(x,y),(x, y) = (0, 0))  does not exist), but   diff(f(x,y),x) = 0, diff(f(x,y),y) = 0if ( x, y ) =  ( 0, 0 ).

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Note that diff(f(x,y),x) = -y*(x^2-y^2)/((x^2+y^2)^2)  and diff(f(x,y),y) = x*(x^2-y^2)/((x^2+y^2)^2)   for all (x, y) other than (0,0).

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Now let's consider the partial derivatives of the function g(x,y) = x^3+x^2*y^3-2*y^2  .

[Maple Plot]g(x,y) = x^3+x^2*y^3-2*y^2
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[Maple Plot]g(x,y) = x^3+x^2*y^3-2*y^2
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diff(g(x,y),x) = 3*x^2+2*x*y^3   and   diff(g(x,y),y) = 3*x^2*y^2-4*y .

[Maple Plot]diff(g(x,y),x) = 3*x^2+2*x*y^3
3DView

[Maple Plot]diff(g(x,y),y) = 3*x^2*y^2-4*y
3DView

diff(f(x,y),`$`(x,2)) = 6*x+2*y^3  ,

diff(diff(f(x,y),x),y) = 6*x*y^2  ,

diff(diff(f(x,y),y),x) = 6*x*y^2  ,

and

diff(f(x,y),`$`(y,2)) = 6*x^2*y-4  .

Here are the graphs of the partial derivatives :

[Maple Plot]diff(f(x,y),`$`(x,2)) = 6*x+2*y^3
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[Maple Plot]diff(diff(f(x,y),x),y) = 6*x*y^2
3DView

[Maple Plot]diff(f(x,y),`$`(y,2)) = 6*x^2*y-4
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Notice that here we have diff(diff(g(x,y),x),y) = diff(diff(g(x,y),y),x) , but this is not true in general.

Let's go back to the function f(x,y) = x*y/(x^2+y^2) for all ( x,y) except at (0,0), and f(0,0) = 0.

Since

[Maple OLE 2.0 Object]    and

[Maple OLE 2.0 Object] .        

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We have seen that   diff(f(x,y),x) = 0, diff(f(x,y),y) = 0 if (x, y) = (0, 0) and

diff(f(x,y),x) = -y*(x^2-y^2)/((x^2+y^2)^2)and diff(f(x,y),y) = x*(x^2-y^2)/((x^2+y^2)^2)

for all (x, y) other than (0, 0). We get

[Maple OLE 2.0 Object]  and [Maple OLE 2.0 Object] .

In fact, we can also compute them directly from the definition :

[Maple OLE 2.0 Object] .

and

[Maple OLE 2.0 Object] .

It is not hard to see that diff(diff(f(x,y),x),y)and diff(diff(f(x,y),y),x)do not exist at ( 0,0).

Let's take a look at the function h(x,y) = (x^3*y-x*y^3)/(x^2+y^2)for all (x,y) except at (0,0),and g(0,0) = 0 .

[Maple Plot]h(x,y) = (x^3*y-x*y^3)/(x^2+y^2)
3DView

Calculations, which we leave as an exercise, show that

[Maple OLE 2.0 Object] and [Maple OLE 2.0 Object] .

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>    diff((x^3*y-x*y^3)/(x^2+y^2),x):
simplify(%);
diff((x^3*y-x*y^3)/(x^2+y^2),y):
simplify(%);

y*(x^4+4*x^2*y^2-y^4)/(x^2+y^2)^2

x*(x^4-4*x^2*y^2-y^4)/(x^2+y^2)^2

Here are the graphs of  diff(h(x,y),x)   and   diff(h(x,y),y) .

[Maple Plot]diff(h(x,y),x)
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[Maple Plot] diff(h(x,y),y)
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>    x:='x':
y:='y':
diff(diff((x^3*y-x*y^3)/(x^2+y^2),y),x):
simplify(%);
diff(diff((x^3*y-x*y^3)/(x^2+y^2),x),y):
simplify(%);

(x^6+9*x^4*y^2-9*x^2*y^4-y^6)/(x^2+y^2)^3

(x^6+9*x^4*y^2-9*x^2*y^4-y^6)/(x^2+y^2)^3

Here are the graphs of   diff(diff(h(x,y),x),y)  and   diff(diff(h(x,y),y),x) .

[Maple Plot]
3DView

[Maple Plot]
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Note that diff(diff(h(x,y),x),y)  and   diff(diff(h(x,y),y),x) are not continuous at ( 0,0).

Clairaut's Theorem

Suppose that f  is defined on a disk D  that contains (a, b). If the functions f[x*y](x,y)  and  f[x*y](x,y) are continuous on D , then f[x*y](a,b) = f[y*x](a,b) .

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