Solve (frac{x^{-2}y^{3}}{x^{2}y})^-1/2(x^3y/y^1/2)^2

$$( \frac { x ^ { - 2 } y ^ { 3 } } { x ^ { 2 } y } ) ^ { - 1 / 2 } ( \frac { x ^ { 3 } y } { y ^ { 1 / 2 } } ) ^ { 2 }$$

Evaluate

$$x^{8}$$

Solution Steps

Cancel out y in both numerator and denominator.

$$\left(\frac{x^{-2}y^{2}}{x^{2}}\right)^{-\frac{1}{2}}\times \left(\frac{x^{3}y}{y^{\frac{1}{2}}}\right)^{2}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\left(\frac{y^{2}}{x^{4}}\right)^{-\frac{1}{2}}\times \left(\frac{x^{3}y}{y^{\frac{1}{2}}}\right)^{2}$$

To raise \frac{y^{2}}{x^{4}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(y^{2}\right)^{-\frac{1}{2}}}{\left(x^{4}\right)^{-\frac{1}{2}}}\times \left(\frac{x^{3}y}{y^{\frac{1}{2}}}\right)^{2}$$

Cancel out \sqrt{y} in both numerator and denominator.

$$\frac{\left(y^{2}\right)^{-\frac{1}{2}}}{\left(x^{4}\right)^{-\frac{1}{2}}}\left(\sqrt{y}x^{3}\right)^{2}$$

Expand \left(\sqrt{y}x^{3}\right)^{2}.

$$\frac{\left(y^{2}\right)^{-\frac{1}{2}}}{\left(x^{4}\right)^{-\frac{1}{2}}}\left(\sqrt{y}\right)^{2}\left(x^{3}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{\left(y^{2}\right)^{-\frac{1}{2}}}{\left(x^{4}\right)^{-\frac{1}{2}}}\left(\sqrt{y}\right)^{2}x^{6}$$

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

$$\frac{\left(y^{2}\right)^{-\frac{1}{2}}}{\left(x^{4}\right)^{-\frac{1}{2}}}yx^{6}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -\frac{1}{2} to get -1.

$$\frac{y^{-1}}{\left(x^{4}\right)^{-\frac{1}{2}}}yx^{6}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -\frac{1}{2} to get -2.

$$\frac{y^{-1}}{x^{-2}}yx^{6}$$

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

$$\frac{y^{-1}y}{x^{-2}}x^{6}$$

Express \frac{y^{-1}y}{x^{-2}}x^{6} as a single fraction.

$$\frac{y^{-1}yx^{6}}{x^{-2}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{1}{y}yx^{8}$$

Cancel out y and y.

$$x^{8}$$

Differentiate w.r.t. x

$$8x^{7}$$

Steps Using Definition of a Derivative

Cancel out y in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{x^{-2}y^{2}}{x^{2}}\right)^{-\frac{1}{2}}\times \left(\frac{x^{3}y}{y^{\frac{1}{2}}}\right)^{2})$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{y^{2}}{x^{4}}\right)^{-\frac{1}{2}}\times \left(\frac{x^{3}y}{y^{\frac{1}{2}}}\right)^{2})$$

To raise \frac{y^{2}}{x^{4}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(y^{2}\right)^{-\frac{1}{2}}}{\left(x^{4}\right)^{-\frac{1}{2}}}\times \left(\frac{x^{3}y}{y^{\frac{1}{2}}}\right)^{2})$$

Cancel out \sqrt{y} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(y^{2}\right)^{-\frac{1}{2}}}{\left(x^{4}\right)^{-\frac{1}{2}}}\left(\sqrt{y}x^{3}\right)^{2})$$

Expand \left(\sqrt{y}x^{3}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(y^{2}\right)^{-\frac{1}{2}}}{\left(x^{4}\right)^{-\frac{1}{2}}}\left(\sqrt{y}\right)^{2}\left(x^{3}\right)^{2})$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(y^{2}\right)^{-\frac{1}{2}}}{\left(x^{4}\right)^{-\frac{1}{2}}}\left(\sqrt{y}\right)^{2}x^{6})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(y^{2}\right)^{-\frac{1}{2}}}{\left(x^{4}\right)^{-\frac{1}{2}}}yx^{6})$$

To raise a power to another power, multiply the exponents. Multiply 2 and -\frac{1}{2} to get -1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y^{-1}}{\left(x^{4}\right)^{-\frac{1}{2}}}yx^{6})$$

To raise a power to another power, multiply the exponents. Multiply 4 and -\frac{1}{2} to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y^{-1}}{x^{-2}}yx^{6})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y^{-1}y}{x^{-2}}x^{6})$$

Express \frac{y^{-1}y}{x^{-2}}x^{6} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y^{-1}yx^{6}}{x^{-2}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{y}yx^{8})$$

Cancel out y and y.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{8})$$

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

$$8x^{8-1}$$

Subtract 1 from 8.

$$8x^{7}$$