let f(x)= arctan((x/y)) calculate (∂^2 f/∂x^2 )(x,y) , (∂^2 f/∂y^2 )(x,y), (∂^2 f/(∂x∂y))(x,y) (∂^2 f/(∂y∂x))(x,y)


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Question Number 36177 by prof Abdo imad last updated on 30/May/18

$l e t f (x) = a r c t a n (\frac{x}{y})$ $c a l c u l a t e \frac{\partial^{2} f}{\partial x^{2}} (x, y), \frac{\partial^{2} f}{\partial y^{2}} (x, y), \frac{\partial^{2} f}{\partial x \partial y} (x, y)$ $\frac{\partial^{2} f}{\partial y \partial x} (x, y)$
Commented by maxmathsup by imad last updated on 19/Aug/18
$we have (∂f/∂x)(x,y) = (1/(y(1+(x^2 /y^2 )))) = (1/(y +(x^2 /y))) = (y/(x^2 +y^2 )) ⇒ (∂^2 f/∂x^2 )(x,y) =(∂/∂x){ (y/(x^2 +y^2 ))} =y.((−2x)/((x^2 +y^2 )^2 )) =((−2xy)/((x^2 +y^2 )^2 )) also we have (∂f/∂y)(x,y) =((−x)/(y^2 (1+(x^2 /y^2 )))) =((−x)/(y^2 +x^2 )) ⇒ (∂^2 f/∂y^2 )(x,y) =(∂/∂y){ ((−x)/(y^2 +x^2 ))} =−x ((−2y)/((y^2 +x^2 )^2 )) =((2xy)/((y^2 +x^2 )^2 )) also (∂^2 f/(∂y∂x))(x,y) =(∂/∂y)((∂f/∂x)(x,y))=(∂/∂y){(y/(x^2 +y^2 ))}=((x^2 +y^2 −y(2y))/((x^2 +y^2 )^2 )) =((x^2 −y^2 )/((x^2 +y^2 )^2 )) also (∂^2 f/(∂x∂y))(x,y) = (∂/∂x)((∂f/∂y)(x,y)) =(∂/∂x){ ((−x)/(x^2 +y^2 ))} =((−(x^2 +y^2 )+x(2x))/((x^2 +y^2 )^2 )) =((x^2 −y^2 )/((x^2 +y^2 )^2 )) .$
$w e h a v e \frac{\partial f}{\partial x} (x, y) = \frac{1}{y (1 + \frac{x^{2}}{y^{2}})} = \frac{1}{y + \frac{x^{2}}{y}} = \frac{y}{x^{2} + y^{2}} \Rightarrow$ $\frac{\partial^{2} f}{\partial x^{2}} (x, y) = \frac{\partial}{\partial x} {\frac{y}{x^{2} + y^{2}}} = y . \frac{- 2 x}{{(x^{2} + y^{2})}^{2}} = \frac{- 2 x y}{{(x^{2} + y^{2})}^{2}} a l s o w e h a v e$ $\frac{\partial f}{\partial y} (x, y) = \frac{- x}{y^{2} (1 + \frac{x^{2}}{y^{2}})} = \frac{- x}{y^{2} + x^{2}} \Rightarrow$ $\frac{\partial^{2} f}{\partial y^{2}} (x, y) = \frac{\partial}{\partial y} {\frac{- x}{y^{2} + x^{2}}} = - x \frac{- 2 y}{{(y^{2} + x^{2})}^{2}} = \frac{2 x y}{{(y^{2} + x^{2})}^{2}} a l s o$ $\frac{\partial^{2} f}{\partial y \partial x} (x, y) = \frac{\partial}{\partial y} (\frac{\partial f}{\partial x} (x, y)) = \frac{\partial}{\partial y} {\frac{y}{x^{2} + y^{2}}} = \frac{x^{2} + y^{2} - y (2 y)}{{(x^{2} + y^{2})}^{2}}$ $= \frac{x^{2} - y^{2}}{{(x^{2} + y^{2})}^{2}} a l s o$ $\frac{\partial^{2} f}{\partial x \partial y} (x, y) = \frac{\partial}{\partial x} (\frac{\partial f}{\partial y} (x, y)) = \frac{\partial}{\partial x} {\frac{- x}{x^{2} + y^{2}}} = \frac{- (x^{2} + y^{2}) + x (2 x)}{{(x^{2} + y^{2})}^{2}}$ $= \frac{x^{2} - y^{2}}{{(x^{2} + y^{2})}^{2}} .$
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