Solve (1)*(-2x^{frac{1/2}*y)(1/2x^{-1/2}*y^frac{2{3}})^2(-4x^frac{1{2}}*y^frac{2{3}})}{y}

$$( 1 ) \cdot \frac { ( - 2 x ^ { \frac { 1 } { 2 } } \cdot y ) ( \frac { 1 } { 2 } x ^ { - \frac { 1 } { 2 } } \cdot y ^ { \frac { 2 } { 3 } } ) ^ { 2 } ( - 4 x ^ { \frac { 1 } { 2 } } \cdot y ^ { \frac { 2 } { 3 } } ) } { y }$$

Evaluate

$$2y^{2}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{1}{2} to get 1.

$$1\times \frac{-2x^{1}y\times \left(\frac{1}{2}x^{-\frac{1}{2}}y^{\frac{2}{3}}\right)^{2}\left(-4\right)y^{\frac{2}{3}}}{y}$$

To multiply powers of the same base, add their exponents. Add 1 and \frac{2}{3} to get \frac{5}{3}.

$$1\times \frac{-2x^{1}y^{\frac{5}{3}}\times \left(\frac{1}{2}x^{-\frac{1}{2}}y^{\frac{2}{3}}\right)^{2}\left(-4\right)}{y}$$

Cancel out y in both numerator and denominator.

$$1\left(-4\right)\left(-2\right)\times \left(\frac{1}{2}x^{-\frac{1}{2}}y^{\frac{2}{3}}\right)^{2}y^{\frac{2}{3}}x^{1}$$

Multiply -4 and -2 to get 8.

$$1\times 8\times \left(\frac{1}{2}x^{-\frac{1}{2}}y^{\frac{2}{3}}\right)^{2}y^{\frac{2}{3}}x^{1}$$

Multiply 1 and 8 to get 8.

$$8\times \left(\frac{1}{2}x^{-\frac{1}{2}}y^{\frac{2}{3}}\right)^{2}y^{\frac{2}{3}}x^{1}$$

Expand \left(\frac{1}{2}x^{-\frac{1}{2}}y^{\frac{2}{3}}\right)^{2}.

$$8\times \left(\frac{1}{2}\right)^{2}\left(x^{-\frac{1}{2}}\right)^{2}\left(y^{\frac{2}{3}}\right)^{2}y^{\frac{2}{3}}x^{1}$$

To raise a power to another power, multiply the exponents. Multiply -\frac{1}{2} and 2 to get -1.

$$8\times \left(\frac{1}{2}\right)^{2}x^{-1}\left(y^{\frac{2}{3}}\right)^{2}y^{\frac{2}{3}}x^{1}$$

To raise a power to another power, multiply the exponents. Multiply \frac{2}{3} and 2 to get \frac{4}{3}.

$$8\times \left(\frac{1}{2}\right)^{2}x^{-1}y^{\frac{4}{3}}y^{\frac{2}{3}}x^{1}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$8\times \frac{1}{4}x^{-1}y^{\frac{4}{3}}y^{\frac{2}{3}}x^{1}$$

Multiply 8 and \frac{1}{4} to get 2.

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

To multiply powers of the same base, add their exponents. Add \frac{4}{3} and \frac{2}{3} to get 2.

$$2x^{-1}y^{2}x^{1}$$

Calculate x to the power of 1 and get x.

$$2x^{-1}y^{2}x$$

Multiply x^{-1} and x to get 1.

$$2y^{2}$$

Differentiate w.r.t. y

$$4y$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{1}{2} to get 1.

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

To multiply powers of the same base, add their exponents. Add 1 and \frac{2}{3} to get \frac{5}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(1\times \frac{-2x^{1}y^{\frac{5}{3}}\times \left(\frac{1}{2}x^{-\frac{1}{2}}y^{\frac{2}{3}}\right)^{2}\left(-4\right)}{y})$$

Cancel out y in both numerator and denominator.

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

Multiply -4 and -2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}y}(1\times 8\times \left(\frac{1}{2}x^{-\frac{1}{2}}y^{\frac{2}{3}}\right)^{2}y^{\frac{2}{3}}x^{1})$$

Multiply 1 and 8 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}y}(8\times \left(\frac{1}{2}x^{-\frac{1}{2}}y^{\frac{2}{3}}\right)^{2}y^{\frac{2}{3}}x^{1})$$

Expand \left(\frac{1}{2}x^{-\frac{1}{2}}y^{\frac{2}{3}}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(8\times \left(\frac{1}{2}\right)^{2}\left(x^{-\frac{1}{2}}\right)^{2}\left(y^{\frac{2}{3}}\right)^{2}y^{\frac{2}{3}}x^{1})$$

To raise a power to another power, multiply the exponents. Multiply -\frac{1}{2} and 2 to get -1.

$$\frac{\mathrm{d}}{\mathrm{d}y}(8\times \left(\frac{1}{2}\right)^{2}x^{-1}\left(y^{\frac{2}{3}}\right)^{2}y^{\frac{2}{3}}x^{1})$$

To raise a power to another power, multiply the exponents. Multiply \frac{2}{3} and 2 to get \frac{4}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(8\times \left(\frac{1}{2}\right)^{2}x^{-1}y^{\frac{4}{3}}y^{\frac{2}{3}}x^{1})$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(8\times \frac{1}{4}x^{-1}y^{\frac{4}{3}}y^{\frac{2}{3}}x^{1})$$

Multiply 8 and \frac{1}{4} to get 2.

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

To multiply powers of the same base, add their exponents. Add \frac{4}{3} and \frac{2}{3} to get 2.

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

Calculate x to the power of 1 and get x.

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

Multiply x^{-1} and x to get 1.

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

The derivative of ax^{n} is nax^{n-1}.

$$2\times 2y^{2-1}$$

Multiply 2 times 2.

$$4y^{2-1}$$

Subtract 1 from 2.

$$4y^{1}$$

For any term t, t^{1}=t.

$$4y$$