Solve x-y/frac{x{y}+y/x}

$$x - \frac { y } { \frac { x } { y } + \frac { y } { x } }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y and x is xy. Multiply \frac{x}{y} times \frac{x}{x}. Multiply \frac{y}{x} times \frac{y}{y}.

$$x-\frac{y}{\frac{xx}{xy}+\frac{yy}{xy}}$$

Since \frac{xx}{xy} and \frac{yy}{xy} have the same denominator, add them by adding their numerators.

$$x-\frac{y}{\frac{xx+yy}{xy}}$$

Do the multiplications in xx+yy.

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

Divide y by \frac{x^{2}+y^{2}}{xy} by multiplying y by the reciprocal of \frac{x^{2}+y^{2}}{xy}.

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

Multiply y and y to get y^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x^{2}+y^{2}}{x^{2}+y^{2}}.

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

Since \frac{x\left(x^{2}+y^{2}\right)}{x^{2}+y^{2}} and \frac{y^{2}x}{x^{2}+y^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x\left(x^{2}+y^{2}\right)-y^{2}x.

$$\frac{x^{3}+xy^{2}-y^{2}x}{x^{2}+y^{2}}$$

Combine like terms in x^{3}+xy^{2}-y^{2}x.

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

Differentiate w.r.t. x

$$\left(\frac{x}{x^{2}+y^{2}}\right)^{2}\left(x^{2}+3y^{2}\right)$$