Solve frac{{y}^{3/7}}{{y}^frac{5{7}}}divfrac{{y}^frac{5{3}}}{{y}^frac{9{7}}}*{y}^frac{4{3}}

$$\frac{ { y }^{ \frac{ 3 }{ 7 } } }{ { y }^{ \frac{ 5 }{ 7 } } } \div \frac { { y }^{ \frac{ 5 }{ 3 } } } { { y }^{ \frac{ 9 }{ 7 } } } \times { y }^{ \frac{ 4 }{ 3 } }$$

Evaluate

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

Steps Using Quotient of Powers Property

Rewrite y^{\frac{5}{7}} as y^{\frac{3}{7}}y^{\frac{2}{7}}. Cancel out y^{\frac{3}{7}} in both numerator and denominator.

$$\frac{\frac{1}{y^{\frac{2}{7}}}}{\frac{y^{\frac{5}{3}}}{y^{\frac{9}{7}}}}y^{\frac{4}{3}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{9}{7} from \frac{5}{3} to get \frac{8}{21}.

$$\frac{\frac{1}{y^{\frac{2}{7}}}}{y^{\frac{8}{21}}}y^{\frac{4}{3}}$$

Express \frac{\frac{1}{y^{\frac{2}{7}}}}{y^{\frac{8}{21}}} as a single fraction.

$$\frac{1}{y^{\frac{2}{7}}y^{\frac{8}{21}}}y^{\frac{4}{3}}$$

To multiply powers of the same base, add their exponents. Add \frac{2}{7} and \frac{8}{21} to get \frac{2}{3}.

$$\frac{1}{y^{\frac{2}{3}}}y^{\frac{4}{3}}$$

Use the rules of exponents to simplify the expression.

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

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

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

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

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

Differentiate w.r.t. y

$$\frac{2}{3\sqrt[3]{y}}$$

Solution Steps

Rewrite y^{\frac{5}{7}} as y^{\frac{3}{7}}y^{\frac{2}{7}}. Cancel out y^{\frac{3}{7}} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\frac{1}{y^{\frac{2}{7}}}}{\frac{y^{\frac{5}{3}}}{y^{\frac{9}{7}}}}y^{\frac{4}{3}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{9}{7} from \frac{5}{3} to get \frac{8}{21}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\frac{1}{y^{\frac{2}{7}}}}{y^{\frac{8}{21}}}y^{\frac{4}{3}})$$

Express \frac{\frac{1}{y^{\frac{2}{7}}}}{y^{\frac{8}{21}}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{y^{\frac{2}{7}}y^{\frac{8}{21}}}y^{\frac{4}{3}})$$

To multiply powers of the same base, add their exponents. Add \frac{2}{7} and \frac{8}{21} to get \frac{2}{3}.

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

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

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

Do the arithmetic.

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

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}.

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

Do the arithmetic.

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