Solve (-3^-3y^-3)^-1

$$( - 3 ^ { - 3 } y ^ { - 3 } ) ^ { - 1 }$$

Evaluate

$$-27y^{3}$$

Solution Steps

Calculate 3 to the power of -3 and get \frac{1}{27}.

$$\left(-\frac{1}{27}y^{-3}\right)^{-1}$$

Expand \left(-\frac{1}{27}y^{-3}\right)^{-1}.

$$\left(-\frac{1}{27}\right)^{-1}\left(y^{-3}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -1 to get 3.

$$\left(-\frac{1}{27}\right)^{-1}y^{3}$$

Calculate -\frac{1}{27} to the power of -1 and get -27.

$$-27y^{3}$$

Differentiate w.r.t. y

$$-81y^{2}$$

Steps Using Definition of a Derivative

Calculate 3 to the power of -3 and get \frac{1}{27}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\left(-\frac{1}{27}y^{-3}\right)^{-1})$$

Expand \left(-\frac{1}{27}y^{-3}\right)^{-1}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\left(-\frac{1}{27}\right)^{-1}\left(y^{-3}\right)^{-1})$$

To raise a power to another power, multiply the exponents. Multiply -3 and -1 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\left(-\frac{1}{27}\right)^{-1}y^{3})$$

Calculate -\frac{1}{27} to the power of -1 and get -27.

$$\frac{\mathrm{d}}{\mathrm{d}y}(-27y^{3})$$

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

$$3\left(-27\right)y^{3-1}$$

Multiply 3 times -27.

$$-81y^{3-1}$$

Subtract 1 from 3.

$$-81y^{2}$$