Solve 1/8x(-4x^{2}y)+(-1,2x)(-5/6x^{2}y)-1/2x^2(-xy)+x(-3x^2y)+xy(-4x^2)-2y(-7x^3)

$$\frac { 1 } { 8 } x ( - 4 x ^ { 2 } y ) + ( - 1,2 x ) ( - \frac { 5 } { 6 } x ^ { 2 } y ) - \frac { 1 } { 2 } x ^ { 2 } ( - x y ) + x ( - 3 x ^ { 2 } y ) + x y ( - 4 x ^ { 2 } ) - 2 y ( - 7 x ^ { 3 } )$$

Evaluate

$$8yx^{3}$$

Solution Steps

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

$$\frac{1}{8}x^{3}\left(-4\right)y-1,2x\left(-\frac{5}{6}\right)x^{2}y-\frac{1}{2}x^{2}\left(-x\right)y+x\left(-3\right)x^{2}y+xy\left(-4\right)x^{2}-2y\left(-7\right)x^{3}$$

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

$$\frac{1}{8}x^{3}\left(-4\right)y-1,2x^{3}\left(-\frac{5}{6}\right)y-\frac{1}{2}x^{2}\left(-x\right)y+x\left(-3\right)x^{2}y+xy\left(-4\right)x^{2}-2y\left(-7\right)x^{3}$$

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

$$\frac{1}{8}x^{3}\left(-4\right)y-1,2x^{3}\left(-\frac{5}{6}\right)y-\frac{1}{2}x^{2}\left(-x\right)y+x^{3}\left(-3\right)y+xy\left(-4\right)x^{2}-2y\left(-7\right)x^{3}$$

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

$$\frac{1}{8}x^{3}\left(-4\right)y-1,2x^{3}\left(-\frac{5}{6}\right)y-\frac{1}{2}x^{2}\left(-x\right)y+x^{3}\left(-3\right)y+x^{3}y\left(-4\right)-2y\left(-7\right)x^{3}$$

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

$$-\frac{1}{2}x^{3}y-1,2x^{3}\left(-\frac{5}{6}\right)y-\frac{1}{2}x^{2}\left(-x\right)y+x^{3}\left(-3\right)y+x^{3}y\left(-4\right)-2y\left(-7\right)x^{3}$$

Multiply -1,2 and -\frac{5}{6} to get 1.

$$-\frac{1}{2}x^{3}y+x^{3}y-\frac{1}{2}x^{2}\left(-x\right)y+x^{3}\left(-3\right)y+x^{3}y\left(-4\right)-2y\left(-7\right)x^{3}$$

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

$$\frac{1}{2}x^{3}y-\frac{1}{2}x^{2}\left(-x\right)y+x^{3}\left(-3\right)y+x^{3}y\left(-4\right)-2y\left(-7\right)x^{3}$$

Combine x^{3}\left(-3\right)y and x^{3}y\left(-4\right) to get -7x^{3}y.

$$\frac{1}{2}x^{3}y-\frac{1}{2}x^{2}\left(-x\right)y-7x^{3}y-2y\left(-7\right)x^{3}$$

Multiply 2 and -7 to get -14.

$$\frac{1}{2}x^{3}y-\frac{1}{2}x^{2}\left(-x\right)y-7x^{3}y-\left(-14yx^{3}\right)$$

The opposite of -14yx^{3} is 14yx^{3}.

$$\frac{1}{2}x^{3}y-\frac{1}{2}x^{2}\left(-x\right)y-7x^{3}y+14yx^{3}$$

Combine -7x^{3}y and 14yx^{3} to get 7x^{3}y.

$$\frac{1}{2}x^{3}y-\frac{1}{2}x^{2}\left(-x\right)y+7x^{3}y$$

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

$$\frac{1}{2}x^{3}y-\frac{1}{2}x^{3}\left(-1\right)y+7x^{3}y$$

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

$$\frac{1}{2}x^{3}y+\frac{1}{2}x^{3}y+7x^{3}y$$

Combine \frac{1}{2}x^{3}y and \frac{1}{2}x^{3}y to get x^{3}y.

$$x^{3}y+7x^{3}y$$

Combine x^{3}y and 7x^{3}y to get 8x^{3}y.

$$8x^{3}y$$

Differentiate w.r.t. x

$$24yx^{2}$$