Solve 3x^3+3x^3+3x^3+6x+1x^2+1{x}^{2}+1{x}^2+2{x}^2-1x-1x-1x-2x-13{x}^3-1{x}^2+1x+1

$$3 { x }^{ 3 } +3 { x }^{ 3 } +3 { x }^{ 3 } +6x+1 { x }^{ 2 } +1 { x }^{ 2 } +1 { x }^{ 2 } +2 { x }^{ 2 } -1x-1x-1x-2x-13 { x }^{ 3 } -1 { x }^{ 2 } +1x+1$$

Evaluate

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

Solution Steps

Combine 3x^{3} and 3x^{3} to get 6x^{3}.

$$6x^{3}+3x^{3}+6x+1x^{2}+1x^{2}+1x^{2}+2x^{2}-x-x-x-2x-13x^{3}-x^{2}+1x+1$$

Combine 6x^{3} and 3x^{3} to get 9x^{3}.

$$9x^{3}+6x+1x^{2}+1x^{2}+1x^{2}+2x^{2}-x-x-x-2x-13x^{3}-x^{2}+1x+1$$

Combine 1x^{2} and 1x^{2} to get 2x^{2}.

$$9x^{3}+6x+2x^{2}+1x^{2}+2x^{2}-x-x-x-2x-13x^{3}-x^{2}+1x+1$$

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

$$9x^{3}+6x+3x^{2}+2x^{2}-x-x-x-2x-13x^{3}-x^{2}+1x+1$$

Combine 3x^{2} and 2x^{2} to get 5x^{2}.

$$9x^{3}+6x+5x^{2}-x-x-x-2x-13x^{3}-x^{2}+1x+1$$

Combine 6x and -x to get 5x.

$$9x^{3}+5x+5x^{2}-x-x-2x-13x^{3}-x^{2}+1x+1$$

Combine 5x and -x to get 4x.

$$9x^{3}+4x+5x^{2}-x-2x-13x^{3}-x^{2}+1x+1$$

Combine 4x and -x to get 3x.

$$9x^{3}+3x+5x^{2}-2x-13x^{3}-x^{2}+1x+1$$

Combine 3x and -2x to get x.

$$9x^{3}+x+5x^{2}-13x^{3}-x^{2}+1x+1$$

Combine 9x^{3} and -13x^{3} to get -4x^{3}.

$$-4x^{3}+x+5x^{2}-x^{2}+1x+1$$

Combine 5x^{2} and -x^{2} to get 4x^{2}.

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

Combine x and 1x to get 2x.

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

Differentiate w.r.t. x

$$2+8x-12x^{2}$$

Steps Using Derivative Rule for Sum

Combine 3x^{3} and 3x^{3} to get 6x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{3}+3x^{3}+6x+1x^{2}+1x^{2}+1x^{2}+2x^{2}-x-x-x-2x-13x^{3}-x^{2}+1x+1)$$

Combine 6x^{3} and 3x^{3} to get 9x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(9x^{3}+6x+1x^{2}+1x^{2}+1x^{2}+2x^{2}-x-x-x-2x-13x^{3}-x^{2}+1x+1)$$

Combine 1x^{2} and 1x^{2} to get 2x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(9x^{3}+6x+2x^{2}+1x^{2}+2x^{2}-x-x-x-2x-13x^{3}-x^{2}+1x+1)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(9x^{3}+6x+3x^{2}+2x^{2}-x-x-x-2x-13x^{3}-x^{2}+1x+1)$$

Combine 3x^{2} and 2x^{2} to get 5x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(9x^{3}+6x+5x^{2}-x-x-x-2x-13x^{3}-x^{2}+1x+1)$$

Combine 6x and -x to get 5x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(9x^{3}+5x+5x^{2}-x-x-2x-13x^{3}-x^{2}+1x+1)$$

Combine 5x and -x to get 4x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(9x^{3}+4x+5x^{2}-x-2x-13x^{3}-x^{2}+1x+1)$$

Combine 4x and -x to get 3x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(9x^{3}+3x+5x^{2}-2x-13x^{3}-x^{2}+1x+1)$$

Combine 3x and -2x to get x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(9x^{3}+x+5x^{2}-13x^{3}-x^{2}+1x+1)$$

Combine 9x^{3} and -13x^{3} to get -4x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-4x^{3}+x+5x^{2}-x^{2}+1x+1)$$

Combine 5x^{2} and -x^{2} to get 4x^{2}.

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

Combine x and 1x to get 2x.

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

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}.

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

Multiply 3 times -4.

$$-12x^{3-1}+2x^{1-1}+2\times 4x^{2-1}$$

Subtract 1 from 3.

$$-12x^{2}+2x^{1-1}+2\times 4x^{2-1}$$

Subtract 1 from 1.

$$-12x^{2}+2x^{0}+2\times 4x^{2-1}$$

Multiply 1 times 2.

$$-12x^{2}+2x^{0}+8x^{2-1}$$

Subtract 1 from 2.

$$-12x^{2}+2x^{0}+8x^{1}$$

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

$$-12x^{2}+2x^{0}+8x$$

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

$$-12x^{2}+2\times 1+8x$$

For any term t, t\times 1=t and 1t=t.

$$-12x^{2}+2+8x$$