Solve frac{y^frac{4{7}}}{y^frac{2{3}}}

$$\frac { y ^ { \frac { 4 } { 7 } } } { y ^ { \frac { 2 } { 3 } } }$$

Evaluate

$$\frac{1}{y^{\frac{2}{21}}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{y^{\frac{4}{7}}}{y^{\frac{2}{3}}}$$

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

$$y^{\frac{4}{7}-\frac{2}{3}}$$

Subtract \frac{2}{3} from \frac{4}{7} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{-\frac{2}{21}}$$

Differentiate w.r.t. y

$$-\frac{2}{21y^{\frac{23}{21}}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{1}y^{\frac{4}{7}-\frac{2}{3}})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y^{-\frac{2}{21}})$$

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}.

$$-\frac{2}{21}y^{-\frac{2}{21}-1}$$

Do the arithmetic.

$$-\frac{2}{21}y^{-\frac{23}{21}}$$