Solve xy(1/x)^10(1/y)^10

$$x y ( \frac { 1 } { x } ) ^ { 10 } ( \frac { 1 } { y } ) ^ { 10 }$$

Evaluate

$$\frac{1}{\left(xy\right)^{9}}$$

Solution Steps

To raise \frac{1}{x} to a power, raise both numerator and denominator to the power and then divide.

$$xy\times \frac{1^{10}}{x^{10}}\times \left(\frac{1}{y}\right)^{10}$$

To raise \frac{1}{y} to a power, raise both numerator and denominator to the power and then divide.

$$xy\times \frac{1^{10}}{x^{10}}\times \frac{1^{10}}{y^{10}}$$

Express x\times \frac{1^{10}}{x^{10}} as a single fraction.

$$\frac{x\times 1^{10}}{x^{10}}y\times \frac{1^{10}}{y^{10}}$$

Cancel out x in both numerator and denominator.

$$\frac{1^{10}}{x^{9}}y\times \frac{1^{10}}{y^{10}}$$

Multiply \frac{1^{10}}{x^{9}} times \frac{1^{10}}{y^{10}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1^{10}\times 1^{10}}{x^{9}y^{10}}y$$

To multiply powers of the same base, add their exponents. Add 10 and 10 to get 20.

$$\frac{1^{20}}{x^{9}y^{10}}y$$

Calculate 1 to the power of 20 and get 1.

$$\frac{1}{x^{9}y^{10}}y$$

Express \frac{1}{x^{9}y^{10}}y as a single fraction.

$$\frac{y}{x^{9}y^{10}}$$

Cancel out y in both numerator and denominator.

$$\frac{1}{x^{9}y^{9}}$$

Differentiate w.r.t. x

$$-\frac{9}{y^{9}x^{10}}$$