Solve frac{3^{-1}y^-3(-3^2)^0}{3^3}

$$\frac { 3 ^ { - 1 } y ^ { - 3 } ( - 3 ^ { 2 } ) ^ { 0 } } { 3 ^ { 3 } }$$

Evaluate

$$\frac{1}{81y^{3}}$$

Solution Steps

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

$$\frac{\left(-3^{2}\right)^{0}y^{-3}}{3^{4}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(-9\right)^{0}y^{-3}}{3^{4}}$$

Calculate -9 to the power of 0 and get 1.

$$\frac{1y^{-3}}{3^{4}}$$

Calculate 3 to the power of 4 and get 81.

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

For any term t, t\times 1=t and 1t=t.

$$\frac{y^{-3}}{81}$$

Differentiate w.r.t. y

$$-\frac{1}{27y^{4}}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\left(-3^{2}\right)^{0}y^{-3}}{3^{4}})$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\left(-9\right)^{0}y^{-3}}{3^{4}})$$

Calculate -9 to the power of 0 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1y^{-3}}{3^{4}})$$

Calculate 3 to the power of 4 and get 81.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1y^{-3}}{81})$$

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

$$-3\times \frac{1}{81}y^{-3-1}$$

Multiply -3 times \frac{1}{81}.

$$-\frac{1}{27}y^{-3-1}$$

Subtract 1 from -3.

$$-\frac{1}{27}y^{-4}$$