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

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

Evaluate

$$27y^{9}$$

Solution Steps

To raise \frac{1}{3y^{3}} to a power, raise both numerator and denominator to the power and then divide.

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

Calculate 1 to the power of -3 and get 1.

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

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

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

To raise a power to another power, multiply the exponents. Multiply 3 and -3 to get -9.

$$\frac{1}{3^{-3}y^{-9}}$$

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

$$\frac{1}{\frac{1}{27}y^{-9}}$$

Differentiate w.r.t. y

$$243y^{8}$$

Steps Using Chain Rule

To raise \frac{1}{3y^{3}} to a power, raise both numerator and denominator to the power and then divide.

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

Calculate 1 to the power of -3 and get 1.

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

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

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

To raise a power to another power, multiply the exponents. Multiply 3 and -3 to get -9.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{3^{-3}y^{-9}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{\frac{1}{27}y^{-9}})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(\frac{1}{27}y^{-9}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{27}y^{-9})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-\left(\frac{1}{27}y^{-9}\right)^{-2}\left(-9\right)\times \frac{1}{27}y^{-9-1}$$

Simplify.

$$\frac{1}{3}y^{-10}\times \left(\frac{1}{27}y^{-9}\right)^{-2}$$