Solve (8/y^6)^-frac{1{3}}

$$( \frac { 8 } { y ^ { 6 } } ) ^ { - \frac { 1 } { 3 } }$$

Evaluate

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

Steps for Raising Quotient to a Power

Use the rules of exponents to simplify the expression.

$$\left(\frac{8}{y^{6}}\right)^{-\frac{1}{3}}$$

To raise the quotient of two numbers to a power, raise each number to the power and then divide.

$$\frac{8^{-\frac{1}{3}}}{\left(y^{6}\right)^{-\frac{1}{3}}}$$

To raise a power to another power, multiply the exponents.

$$\frac{\frac{1}{2}}{y^{6\left(-\frac{1}{3}\right)}}$$

Multiply 6 times -\frac{1}{3}.

$$\frac{\frac{1}{2}y^{2}}{1}$$

Differentiate w.r.t. y

$$y$$

Steps Using Chain Rule

To raise \frac{8}{y^{6}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{8^{-\frac{1}{3}}}{\left(y^{6}\right)^{-\frac{1}{3}}})$$

To raise a power to another power, multiply the exponents. Multiply 6 and -\frac{1}{3} to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{8^{-\frac{1}{3}}}{y^{-2}})$$

Calculate 8 to the power of -\frac{1}{3} and get \frac{1}{2}.

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

Express \frac{\frac{1}{2}}{y^{-2}} as a single fraction.

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

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).

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

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}.

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

Simplify.

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