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f-x-y-x-y-2-x-2-y-2-f-u-f-v-x-uv-y-u-v-

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Question Number 424 by 123456 last updated on 25/Jan/15
f(x,y)=(((x+y)^2 )/(x^2 +y^2 )) { (((∂f/∂u)=?)),(((∂f/∂v)=?)) :} { ((x=uv)),((y=u+v)) :}
f(x,y)=(x+y)2x2+y2{∂f∂u=?∂f∂v=?{x=uvy=u+v
Answered by prakash jain last updated on 02/Jan/15
f(u,v)=1+((2xy)/(x^2 +y^2 )) = 1+((2u^2 v+2uv^2 )/((uv)^2 +(u+v)^2 )) (∂f/∂u)=(([(uv)^2 +(u+v)^2 ][4uv+2v^2 ]−(2u^2 v+2uv^2 )(2uv+2u+2v))/([(uv)^2 +(u+v)^2 ]^2 )) Simplification 4u^3 v^3 +4u^3 v+8u^2 v^2 +4uv^3 +2u^2 v^4 +2u^2 v^2 +4uv^3 +2v^4 −4u^3 v^2 −4u^2 v^3 −4u^3 v−4u^2 v^2 −4u^2 v^2 −4uv^3 =4u^3 v^3 +2u^2 v^4 +2u^2 v^2 +4uv^3 +2v^4 −4u^3 v^2 −4u^2 v^3 (∂f/∂u)=((4u^3 v^3 +2u^2 v^4 +2u^2 v^2 +4uv^3 +2v^4 −4u^3 v^2 −4u^2 v^3 )/([(uv)^2 +(u+v)^2 ]^2 )) (∂f/∂v)=((4u^3 v^3 +2v^2 u^4 +2u^2 v^2 +4vu^3 +2u^4 −4v^3 u^2 −4v^2 u^3 )/([(uv)^2 +(u+v)^2 ]^2 ))
f(u,v)=1+2xyx2+y2=1+2u2v+2uv2(uv)2+(u+v)2∂f∂u=[(uv)2+(u+v)2][4uv+2v2]−(2u2v+2uv2)(2uv+2u+2v)[(uv)2+(u+v)2]2Simplification4u3v3+4u3v+8u2v2+4uv3+2u2v4+2u2v2+4uv3+2v4−4u3v2−4u2v3−4u3v−4u2v2−4u2v2−4uv3=4u3v3+2u2v4+2u2v2+4uv3+2v4−4u3v2−4u2v3∂f∂u=4u3v3+2u2v4+2u2v2+4uv3+2v4−4u3v2−4u2v3[(uv)2+(u+v)2]2∂f∂v=4u3v3+2v2u4+2u2v2+4vu3+2u4−4v3u2−4v2u3[(uv)2+(u+v)2]2

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