Solve frac{12y^frac{1{3}}}{4y^frac{1{2}}}

$$\frac { 12 y ^ { \frac { 1 } { 3 } } } { 4 y ^ { \frac { 1 } { 2 } } }$$

Evaluate

$$\frac{3}{\sqrt[6]{y}}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{12^{1}\sqrt[3]{y}}{4^{1}\sqrt{y}}$$

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

$$\frac{12^{1}y^{\frac{1}{3}-\frac{1}{2}}}{4^{1}}$$

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

$$\frac{12^{1}y^{-\frac{1}{6}}}{4^{1}}$$

Divide 12 by 4.

$$3y^{-\frac{1}{6}}$$

Differentiate w.r.t. y

$$-\frac{1}{2y^{\frac{7}{6}}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{12}{4}y^{\frac{1}{3}-\frac{1}{2}})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}y}(3y^{-\frac{1}{6}})$$

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}.

$$-\frac{1}{6}\times 3y^{-\frac{1}{6}-1}$$

Do the arithmetic.

$$-\frac{1}{2}y^{-\frac{7}{6}}$$