Solve x^3+1x^3+x^2+x+1x^2+x+1

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

Evaluate

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

Solution Steps

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

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

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

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

Combine x and x to get 2x.

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

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Sum

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

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

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

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

Combine x and x to get 2x.

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

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}.

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

Multiply 3 times 2.

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

Subtract 1 from 3.

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

Multiply 2 times 2.

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

Subtract 1 from 2.

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

Subtract 1 from 1.

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

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

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

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

$$6x^{2}+4x+2\times 1$$

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

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