Solve x^3+6-8x+x^2-3x^3-5+3x^2-3x-2{x}^2

$${ x }^{ 3 } +6-8x+ { x }^{ 2 } -3 { x }^{ 3 } -5+3 { x }^{ 2 } -3x-2 { x }^{ 2 }$$

Evaluate

$$1-11x+2x^{2}-2x^{3}$$

Solution Steps

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

$$-2x^{3}+6-8x+x^{2}-5+3x^{2}-3x-2x^{2}$$

Subtract 5 from 6 to get 1.

$$-2x^{3}+1-8x+x^{2}+3x^{2}-3x-2x^{2}$$

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

$$-2x^{3}+1-8x+4x^{2}-3x-2x^{2}$$

Combine -8x and -3x to get -11x.

$$-2x^{3}+1-11x+4x^{2}-2x^{2}$$

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

$$-2x^{3}+1-11x+2x^{2}$$

Differentiate w.r.t. x

$$-6x^{2}+4x-11$$

Steps Using Derivative Rule for Sum

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(-2x^{3}+6-8x+x^{2}-5+3x^{2}-3x-2x^{2})$$

Subtract 5 from 6 to get 1.

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

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

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

Combine -8x and -3x to get -11x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-2x^{3}+1-11x+4x^{2}-2x^{2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(-2x^{3}+1-11x+2x^{2})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$3\left(-2\right)x^{3-1}-11x^{1-1}+2\times 2x^{2-1}$$

Multiply 3 times -2.

$$-6x^{3-1}-11x^{1-1}+2\times 2x^{2-1}$$

Subtract 1 from 3.

$$-6x^{2}-11x^{1-1}+2\times 2x^{2-1}$$

Subtract 1 from 1.

$$-6x^{2}-11x^{0}+2\times 2x^{2-1}$$

Multiply 1 times -11.

$$-6x^{2}-11x^{0}+4x^{2-1}$$

Subtract 1 from 2.

$$-6x^{2}-11x^{0}+4x^{1}$$

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

$$-6x^{2}-11x^{0}+4x$$

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

$$-6x^{2}-11+4x$$