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

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

Evaluate

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

Solution Steps

Divide \frac{x^{-2}y^{-2}}{x^{-2}} by \frac{x^{-2-2}+x^{-4}}{x^{-1}} by multiplying \frac{x^{-2}y^{-2}}{x^{-2}} by the reciprocal of \frac{x^{-2-2}+x^{-4}}{x^{-1}}.

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

Cancel out x^{-2} in both numerator and denominator.

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

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

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

Subtract 2 from -2 to get -4.

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

Combine x^{-4} and x^{-4} to get 2x^{-4}.

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

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

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

To multiply powers of the same base, add their exponents. Add 1 and -4 to get -3.

$$\frac{y^{-2}}{x^{-3}\times 2}$$

Differentiate w.r.t. x

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