Solve xy/x^2-y^2div(1/x-y-1/x+y)

$$\frac { x y } { x ^ { 2 } - y ^ { 2 } } \div ( \frac { 1 } { x - y } - \frac { 1 } { x + y } )$$

Evaluate

$$\frac{x}{2}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-y and x+y is \left(x+y\right)\left(x-y\right). Multiply \frac{1}{x-y} times \frac{x+y}{x+y}. Multiply \frac{1}{x+y} times \frac{x-y}{x-y}.

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

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

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

Do the multiplications in x+y-\left(x-y\right).

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

Combine like terms in x+y-x+y.

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

Divide \frac{xy}{x^{2}-y^{2}} by \frac{2y}{\left(x+y\right)\left(x-y\right)} by multiplying \frac{xy}{x^{2}-y^{2}} by the reciprocal of \frac{2y}{\left(x+y\right)\left(x-y\right)}.

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

Cancel out y in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out \left(x+y\right)\left(x-y\right) in both numerator and denominator.

$$\frac{x}{2}$$

Differentiate w.r.t. x

$$\frac{1}{2} = 0.5$$

Steps Using Definition of a Derivative

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-y and x+y is \left(x+y\right)\left(x-y\right). Multiply \frac{1}{x-y} times \frac{x+y}{x+y}. Multiply \frac{1}{x+y} times \frac{x-y}{x-y}.

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

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

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

Do the multiplications in x+y-\left(x-y\right).

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

Combine like terms in x+y-x+y.

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

Divide \frac{xy}{x^{2}-y^{2}} by \frac{2y}{\left(x+y\right)\left(x-y\right)} by multiplying \frac{xy}{x^{2}-y^{2}} by the reciprocal of \frac{2y}{\left(x+y\right)\left(x-y\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{xy\left(x+y\right)\left(x-y\right)}{\left(x^{2}-y^{2}\right)\times 2y})$$

Cancel out y in both numerator and denominator.

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

Factor the expressions that are not already factored in \frac{x\left(x+y\right)\left(x-y\right)}{2\left(x^{2}-y^{2}\right)}.

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

Cancel out \left(x+y\right)\left(x-y\right) in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x}{2})$$

The derivative of ax^{n} is nax^{n-1}.

$$\frac{1}{2}x^{1-1}$$

Subtract 1 from 1.

$$\frac{1}{2}x^{0}$$

For any term t except 0, t^{0}=1.

$$\frac{1}{2}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{1}{2}$$