Solve x^frac{1{3}}divx^frac{2{3}}

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

Evaluate

$$\frac{1}{\sqrt[3]{x}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

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

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

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

Subtract \frac{2}{3} from \frac{1}{3} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

Differentiate w.r.t. x

$$-\frac{1}{3x^{\frac{4}{3}}}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

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

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$-\frac{1}{3}x^{-\frac{4}{3}}$$