Solve frac{s^frac{1{4}}}{s^frac{7{4}}}

$$\frac { s ^ { \frac { 1 } { 4 } } } { s ^ { \frac { 7 } { 4 } } }$$

Evaluate

$$\frac{1}{s^{\frac{3}{2}}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{\sqrt[4]{s}}{s^{\frac{7}{4}}}$$

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

$$s^{\frac{1}{4}-\frac{7}{4}}$$

Subtract \frac{7}{4} from \frac{1}{4} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$s^{-\frac{3}{2}}$$

Differentiate w.r.t. s

$$-\frac{3}{2s^{\frac{5}{2}}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}s}(\frac{1}{1}s^{\frac{1}{4}-\frac{7}{4}})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$-\frac{3}{2}s^{-\frac{5}{2}}$$