#### Answer

$\Big(\dfrac{x^{2}-y^{2}}{x^{2}+y^{2}}\div\dfrac{x^{2}-y^{2}}{3x}\Big)\cdot\dfrac{x^{2}+y^{2}}{6}=\dfrac{x}{2}$

#### Work Step by Step

$\Big(\dfrac{x^{2}-y^{2}}{x^{2}+y^{2}}\div\dfrac{x^{2}-y^{2}}{3x}\Big)\cdot\dfrac{x^{2}+y^{2}}{6}$
Factor the numerators of the two rational expressions inside the parentheses:
$\Big(\dfrac{(x+y)(x-y)}{x^{2}+y^{2}}\div\dfrac{(x-y)(x+y)}{3x}\Big)\cdot\dfrac{x^{2}+y^{2}}{6}=...$
Evaluate the division of the two rational expressions inside the parentheses:
$...=\dfrac{3x(x+y)(x-y)}{(x^{2}+y^{2})(x-y)(x+y)}\cdot\dfrac{x^{2}+y^{2}}{6}=...$
Now, evaluate the product and simplify by removing the factors that appear both in the numerator and the denominator of the resulting expression:
$...=\dfrac{3x(x^{2}+y^{2})(x+y)(x-y)}{6(x^{2}+y^{2})(x-y)(x+y)}=\dfrac{3x}{6}=\dfrac{x}{2}$