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

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

Evaluate

$$y^{3}x^{4}$$

Solution Steps

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

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

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

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

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

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

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

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

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

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

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

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

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

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

Combine -\frac{1}{2}x^{4}y^{3} and x^{4}y^{3}\left(-\frac{3}{2}\right) to get -2x^{4}y^{3}.

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

Multiply -1 and -1 to get 1.

$$-2x^{4}y^{3}+x^{4}y^{3}+2x^{4}y^{3}$$

Combine -2x^{4}y^{3} and x^{4}y^{3} to get -x^{4}y^{3}.

$$-x^{4}y^{3}+2x^{4}y^{3}$$

Combine -x^{4}y^{3} and 2x^{4}y^{3} to get x^{4}y^{3}.

$$x^{4}y^{3}$$

Differentiate w.r.t. x

$$4\left(xy\right)^{3}$$