# Find Fxx of the following equation?

## f(x, y) = 5x arctan(x/y) I found Fx of this but I have trouble finding Fxx

May 24, 2018

$\frac{10 {y}^{3}}{{x}^{2} + {y}^{2}} ^ 2$

#### Explanation:

$f \left(x , y\right) = 5 x {\tan}^{-} 1 \left(\frac{x}{y}\right) \implies$

${f}_{x} \left(x , y\right) = 5 {\tan}^{-} 1 \left(\frac{x}{y}\right) + 5 x \times \frac{1}{1 + {\left(\frac{x}{y}\right)}^{2}} \times \frac{1}{y}$
$q \quad = 5 {\tan}^{-} 1 \left(\frac{x}{y}\right) + 5 \frac{x y}{{x}^{2} + {y}^{2}} \implies$

${f}_{x x} \left(x , y\right) = 5 \times \frac{1}{1 + {\left(\frac{x}{y}\right)}^{2}} \times \frac{1}{y}$
$q \quad q \quad q \quad q \quad + 5 \frac{\left({x}^{2} + {y}^{2}\right) \times y - x y \times 2 x}{{x}^{2} + {y}^{2}} ^ 2$

$q \quad q \quad = \frac{5 y}{{x}^{2} + {y}^{2}} + \frac{5 y \left({y}^{2} - {x}^{2}\right)}{{x}^{2} + {y}^{2}} ^ 2$
$q \quad q \quad = \frac{5 y \left\{\left({x}^{2} + {y}^{2}\right) + \left({y}^{2} - {x}^{2}\right)\right\}}{{x}^{2} + {y}^{2}} ^ 2$
$q \quad q \quad = \frac{10 {y}^{3}}{{x}^{2} + {y}^{2}} ^ 2$