Solve (frac{x^{-1/3}x^1/6}{x^1/4x^-1/2})^-1/3

$$( \frac { x ^ { - 1 / 3 } x ^ { 1 / 6 } } { x ^ { 1 / 4 } x ^ { - 1 / 2 } } ) ^ { - 1 / 3 }$$

Differentiate w.r.t. x

$$-\frac{1}{36x^{\frac{37}{36}}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add -\frac{1}{3} and \frac{1}{6} to get -\frac{1}{6}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{x^{-\frac{1}{6}}}{x^{\frac{1}{4}}x^{-\frac{1}{2}}}\right)^{-\frac{1}{3}})$$

To multiply powers of the same base, add their exponents. Add \frac{1}{4} and -\frac{1}{2} to get -\frac{1}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{x^{-\frac{1}{6}}}{x^{-\frac{1}{4}}}\right)^{-\frac{1}{3}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -\frac{1}{4} from -\frac{1}{6} to get \frac{1}{12}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(x^{\frac{1}{12}}\right)^{-\frac{1}{3}})$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{12} and -\frac{1}{3} to get -\frac{1}{36}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{-\frac{1}{36}})$$

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

$$-\frac{1}{36}x^{-\frac{1}{36}-1}$$

Subtract 1 from -\frac{1}{36}.

$$-\frac{1}{36}x^{-\frac{37}{36}}$$

Evaluate

$$\frac{1}{\sqrt[36]{x}}$$