Solve (1/216x^3)^-frac{1{3}}

$$( \frac { 1 } { 216 x ^ { 3 } } ) ^ { - \frac { 1 } { 3 } }$$

Evaluate

$$6x$$

Solution Steps

To raise \frac{1}{216x^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1^{-\frac{1}{3}}}{\left(216x^{3}\right)^{-\frac{1}{3}}}$$

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

$$\frac{1}{\left(216x^{3}\right)^{-\frac{1}{3}}}$$

Expand \left(216x^{3}\right)^{-\frac{1}{3}}.

$$\frac{1}{216^{-\frac{1}{3}}\left(x^{3}\right)^{-\frac{1}{3}}}$$

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

$$\frac{1}{216^{-\frac{1}{3}}x^{-1}}$$

Calculate 216 to the power of -\frac{1}{3} and get \frac{1}{6}.

$$\frac{1}{\frac{1}{6}x^{-1}}$$

Differentiate w.r.t. x

$$6$$

Steps Using Chain Rule

To raise \frac{1}{216x^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1^{-\frac{1}{3}}}{\left(216x^{3}\right)^{-\frac{1}{3}}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{\left(216x^{3}\right)^{-\frac{1}{3}}})$$

Expand \left(216x^{3}\right)^{-\frac{1}{3}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{216^{-\frac{1}{3}}\left(x^{3}\right)^{-\frac{1}{3}}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{216^{-\frac{1}{3}}x^{-1}})$$

Calculate 216 to the power of -\frac{1}{3} and get \frac{1}{6}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{\frac{1}{6}x^{-1}})$$

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).

$$-\left(\frac{1}{6}\times \frac{1}{x}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{6}\times \frac{1}{x})$$

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}.

$$-\left(\frac{1}{6}\times \frac{1}{x}\right)^{-2}\left(-1\right)\times \frac{1}{6}x^{-1-1}$$

Simplify.

$$\frac{1}{6}x^{-2}\times \left(\frac{1}{6}\times \frac{1}{x}\right)^{-2}$$