If-z-x-2-tan-1-y-x-find-2-z-x-y-at-1-1-

Question Number 116319 by bemath last updated on 03/Oct/20

If z = x^{2} \tan^{- 1} (\frac{y}{x}), find \frac{\partial^{2} z}{\partial x \partial y}

at (1, 1)

Answered by john santu last updated on 03/Oct/20

\frac{\partial z}{\partial y} = x^{2} . \frac{x^{2}}{x^{2} + y^{2}} . \frac{\partial}{\partial y} (\frac{y}{x}) = \frac{x^{3}}{x^{2} + y^{2}}

\frac{\partial^{2} z}{\partial x \partial y} = \frac{\partial}{\partial x} (\frac{\partial z}{\partial y}) = \frac{\partial}{\partial x} (\frac{x^{3}}{x^{2} + y^{2}})

= \frac{3 x^{2} (x^{2} + y^{2}) - x^{3} (2 x)}{{(x^{2} + y^{2})}^{2}}

= \frac{x^{4} + 3 x^{2} y^{2}}{{(x^{2} + y^{2})}^{2}} ∣_{(1, 1)} = \frac{1 + 3}{{(1 + 1)}^{2}}

= 1 .