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

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

Evaluate

$$\frac{1}{x}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

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

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

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

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

$$\frac{1}{x}$$

Differentiate w.r.t. x

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

Solution Steps

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

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

Do the arithmetic.

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

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}.

$$-x^{-1-1}$$

Do the arithmetic.

$$-x^{-2}$$