Solve frac{sqrt{x^{2}+y^{2}}+x}{y+sqrt{x^{2}-y^{2}}}*frac{x-sqrt{x^2+y^2}}{sqrt{x^2-y^2}-y}

$$\frac { \sqrt { x ^ { 2 } + y ^ { 2 } } + x } { y + \sqrt { x ^ { 2 } - y ^ { 2 } } } \times \frac { x - \sqrt { x ^ { 2 } + y ^ { 2 } } } { \sqrt { x ^ { 2 } - y ^ { 2 } } - y }$$

Evaluate

$$-\frac{y^{2}}{x^{2}-2y^{2}}$$

Short Solution Steps

Multiply \frac{\sqrt{x^{2}+y^{2}}+x}{y+\sqrt{x^{2}-y^{2}}} times \frac{x-\sqrt{x^{2}+y^{2}}}{\sqrt{x^{2}-y^{2}}-y} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(\sqrt{x^{2}+y^{2}}+x\right)\left(x-\sqrt{x^{2}+y^{2}}\right)}{\left(y+\sqrt{x^{2}-y^{2}}\right)\left(\sqrt{x^{2}-y^{2}}-y\right)}$$

Consider \left(\sqrt{x^{2}+y^{2}}+x\right)\left(x-\sqrt{x^{2}+y^{2}}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{x^{2}-\left(\sqrt{x^{2}+y^{2}}\right)^{2}}{\left(y+\sqrt{x^{2}-y^{2}}\right)\left(\sqrt{x^{2}-y^{2}}-y\right)}$$

Calculate \sqrt{x^{2}+y^{2}} to the power of 2 and get x^{2}+y^{2}.

$$\frac{x^{2}-\left(x^{2}+y^{2}\right)}{\left(y+\sqrt{x^{2}-y^{2}}\right)\left(\sqrt{x^{2}-y^{2}}-y\right)}$$

Consider \left(y+\sqrt{x^{2}-y^{2}}\right)\left(\sqrt{x^{2}-y^{2}}-y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{x^{2}-\left(x^{2}+y^{2}\right)}{\left(\sqrt{x^{2}-y^{2}}\right)^{2}-y^{2}}$$

Calculate \sqrt{x^{2}-y^{2}} to the power of 2 and get x^{2}-y^{2}.

$$\frac{x^{2}-\left(x^{2}+y^{2}\right)}{x^{2}-y^{2}-y^{2}}$$

Combine -y^{2} and -y^{2} to get -2y^{2}.

$$\frac{x^{2}-\left(x^{2}+y^{2}\right)}{x^{2}-2y^{2}}$$

To find the opposite of x^{2}+y^{2}, find the opposite of each term.

$$\frac{x^{2}-x^{2}-y^{2}}{x^{2}-2y^{2}}$$

Combine x^{2} and -x^{2} to get 0.

$$\frac{-y^{2}}{x^{2}-2y^{2}}$$

Differentiate w.r.t. x

$$2\times \left(\frac{y}{x^{2}-2y^{2}}\right)^{2}x$$