Solve (-2x)(-xy)^-1

$$( - 2 x ) ( - x y ) ^ { - 1 }$$

Evaluate

$$\frac{2}{y}$$

Solution Steps

Expand \left(\left(-x\right)y\right)^{-1}.

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

Expand \left(-x\right)^{-1}.

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

Calculate -1 to the power of -1 and get -1.

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

Multiply -2 and -1 to get 2.

$$2xx^{-1}y^{-1}$$

Multiply x and x^{-1} to get 1.

$$2y^{-1}$$

Differentiate w.r.t. y

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

Steps Using Definition of a Derivative

Expand \left(\left(-x\right)y\right)^{-1}.

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

Expand \left(-x\right)^{-1}.

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

Calculate -1 to the power of -1 and get -1.

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

Multiply -2 and -1 to get 2.

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

Multiply x and x^{-1} to get 1.

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

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

$$-2y^{-1-1}$$

Subtract 1 from -1.

$$-2y^{-2}$$