Solve -3x^{2}y:(-3xy)-[x^{2}-(x^{2}-1)-xy-1]*[2x^{2}-2x-(-4x^{4}y^2):(-2x^2y^2)]+2x^2y

$$- 3 x ^ { 2 } y : ( - 3 x y ) - [ x ^ { 2 } - ( x ^ { 2 } - 1 ) - x y - 1 ] \cdot [ 2 x ^ { 2 } - 2 x - ( - 4 x ^ { 4 } y ^ { 2 } ) : ( - 2 x ^ { 2 } y ^ { 2 } ) ] + 2 x ^ { 2 } y$$

Evaluate

$$x$$

Solution Steps

Cancel out -3xy in both numerator and denominator.

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

Cancel out 2x^{2}y^{2} in both numerator and denominator.

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

Anything divided by -1 gives its opposite.

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

Subtract 2x^{2} from 2x^{2} to get 0.

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

To find the opposite of x^{2}-1, find the opposite of each term.

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

Combine x^{2} and -x^{2} to get 0.

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

Subtract 1 from 1 to get 0.

$$x-\left(-xy\left(-2\right)x\right)+2x^{2}y$$

Multiply -1 and -2 to get 2.

$$x-2xyx+2x^{2}y$$

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

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

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

$$x$$

Differentiate w.r.t. x

$$1$$

Steps Using Definition of a Derivative

Cancel out -3xy in both numerator and denominator.

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

Cancel out 2x^{2}y^{2} in both numerator and denominator.

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

Anything divided by -1 gives its opposite.

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

Subtract 2x^{2} from 2x^{2} to get 0.

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

To find the opposite of x^{2}-1, find the opposite of each term.

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

Combine x^{2} and -x^{2} to get 0.

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

Subtract 1 from 1 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x-\left(-xy\left(-2\right)x\right)+2x^{2}y)$$

Multiply -1 and -2 to get 2.

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

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

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

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x)$$

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

$$x^{1-1}$$

Subtract 1 from 1.

$$x^{0}$$

For any term t except 0, t^{0}=1.

$$1$$