Solve 2/3xy*(-2x^2y)-(-1/6x^3)*y^2+(-1/2xy^2)3x+3/4{x}^2y*(2y)

$$\frac{ 2 }{ 3 } xy \times (-2 { x }^{ 2 } y)-(- \frac{ 1 }{ 6 } { x }^{ 3 } ) \times { y }^{ 2 } +(- \frac{ 1 }{ 2 } x { y }^{ 2 } )3x+ \frac{ 3 }{ 4 } { x }^{ 2 } y \times (2y)$$

Evaluate

$$-\frac{7y^{2}x^{3}}{6}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{2}{3}x^{3}y\left(-2\right)y-\left(-\frac{1}{6}x^{3}y^{2}\right)-\frac{1}{2}xy^{2}\times 3x+\frac{3}{4}x^{2}y\times 2y$$

Multiply y and y to get y^{2}.

$$\frac{2}{3}x^{3}y^{2}\left(-2\right)-\left(-\frac{1}{6}x^{3}y^{2}\right)-\frac{1}{2}xy^{2}\times 3x+\frac{3}{4}x^{2}y\times 2y$$

Multiply x and x to get x^{2}.

$$\frac{2}{3}x^{3}y^{2}\left(-2\right)-\left(-\frac{1}{6}x^{3}y^{2}\right)-\frac{1}{2}x^{2}y^{2}\times 3+\frac{3}{4}x^{2}y\times 2y$$

Multiply y and y to get y^{2}.

$$\frac{2}{3}x^{3}y^{2}\left(-2\right)-\left(-\frac{1}{6}x^{3}y^{2}\right)-\frac{1}{2}x^{2}y^{2}\times 3+\frac{3}{4}x^{2}y^{2}\times 2$$

Multiply \frac{2}{3} and -2 to get -\frac{4}{3}.

$$-\frac{4}{3}x^{3}y^{2}-\left(-\frac{1}{6}x^{3}y^{2}\right)-\frac{1}{2}x^{2}y^{2}\times 3+\frac{3}{4}x^{2}y^{2}\times 2$$

The opposite of -\frac{1}{6}x^{3}y^{2} is \frac{1}{6}x^{3}y^{2}.

$$-\frac{4}{3}x^{3}y^{2}+\frac{1}{6}x^{3}y^{2}-\frac{1}{2}x^{2}y^{2}\times 3+\frac{3}{4}x^{2}y^{2}\times 2$$

Combine -\frac{4}{3}x^{3}y^{2} and \frac{1}{6}x^{3}y^{2} to get -\frac{7}{6}x^{3}y^{2}.

$$-\frac{7}{6}x^{3}y^{2}-\frac{1}{2}x^{2}y^{2}\times 3+\frac{3}{4}x^{2}y^{2}\times 2$$

Multiply -\frac{1}{2} and 3 to get -\frac{3}{2}.

$$-\frac{7}{6}x^{3}y^{2}-\frac{3}{2}x^{2}y^{2}+\frac{3}{4}x^{2}y^{2}\times 2$$

Multiply \frac{3}{4} and 2 to get \frac{3}{2}.

$$-\frac{7}{6}x^{3}y^{2}-\frac{3}{2}x^{2}y^{2}+\frac{3}{2}x^{2}y^{2}$$

Combine -\frac{3}{2}x^{2}y^{2} and \frac{3}{2}x^{2}y^{2} to get 0.

$$-\frac{7}{6}x^{3}y^{2}$$

Differentiate w.r.t. x

$$-\frac{7\left(xy\right)^{2}}{2}$$