Solve (x^frac{2{3}})^frac{1{2}}*x/x^frac{2{3}}

$$( x ^ { \frac { 2 } { 3 } } ) ^ { \frac { 1 } { 2 } } \times \frac { x } { x ^ { \frac { 2 } { 3 } } }$$

Evaluate

$$\sqrt[3]{x|x|}$$

Solution Steps

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

$$x^{\frac{1}{3}}\times \frac{x}{x^{\frac{2}{3}}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{2}{3} from 1 to get \frac{1}{3}.

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

To multiply powers of the same base, add their exponents. Add \frac{1}{3} and \frac{1}{3} to get \frac{2}{3}.

$$x^{\frac{2}{3}}$$

Differentiate w.r.t. x

$$\frac{2\sqrt[3]{|x|}}{3x^{\frac{2}{3}}}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{1}{3}}\times \frac{x}{x^{\frac{2}{3}}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{2}{3} from 1 to get \frac{1}{3}.

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

To multiply powers of the same base, add their exponents. Add \frac{1}{3} and \frac{1}{3} to get \frac{2}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{2}{3}})$$

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

$$\frac{2}{3}x^{\frac{2}{3}-1}$$

Subtract 1 from \frac{2}{3}.

$$\frac{2}{3}x^{-\frac{1}{3}}$$