Solve 4/x^2/frac{2{x}-1/y}

$$\frac { \frac { 4 } { x ^ { 2 } } } { \frac { 2 } { x } - \frac { 1 } { y } }$$

Evaluate

$$\frac{4y}{x\left(2y-x\right)}$$

Short Solution Steps

Express \frac{\frac{4}{x^{2}}}{\frac{2}{x}-\frac{1}{y}} as a single fraction.

$$\frac{4}{x^{2}\left(\frac{2}{x}-\frac{1}{y}\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and y is xy. Multiply \frac{2}{x} times \frac{y}{y}. Multiply \frac{1}{y} times \frac{x}{x}.

$$\frac{4}{x^{2}\left(\frac{2y}{xy}-\frac{x}{xy}\right)}$$

Since \frac{2y}{xy} and \frac{x}{xy} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4}{x^{2}\times \frac{2y-x}{xy}}$$

Express x^{2}\times \frac{2y-x}{xy} as a single fraction.

$$\frac{4}{\frac{x^{2}\left(2y-x\right)}{xy}}$$

Cancel out x in both numerator and denominator.

$$\frac{4}{\frac{x\left(-x+2y\right)}{y}}$$

Divide 4 by \frac{x\left(-x+2y\right)}{y} by multiplying 4 by the reciprocal of \frac{x\left(-x+2y\right)}{y}.

$$\frac{4y}{x\left(-x+2y\right)}$$

Use the distributive property to multiply x by -x+2y.

$$\frac{4y}{-x^{2}+2xy}$$

Differentiate w.r.t. x

$$\frac{8y\left(x-y\right)}{\left(x\left(2y-x\right)\right)^{2}}$$