Solve (2x^2-2x+6)+(11x^3-x^2-2+5x)

$$(2 { x }^{ 2 } -2x+6)+(11 { x }^{ 3 } - { x }^{ 2 } -2+5x)$$

Evaluate

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

Solution Steps

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

$$x^{2}-2x+6+11x^{3}-2+5x$$

Subtract 2 from 6 to get 4.

$$x^{2}-2x+4+11x^{3}+5x$$

Combine -2x and 5x to get 3x.

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

Differentiate w.r.t. x

$$33x^{2}+2x+3$$

Steps Using Derivative Rule for Sum

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-2x+6+11x^{3}-2+5x)$$

Subtract 2 from 6 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-2x+4+11x^{3}+5x)$$

Combine -2x and 5x to get 3x.

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

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}.

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

Subtract 1 from 2.

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

Subtract 1 from 1.

$$2x^{1}+3x^{0}+3\times 11x^{3-1}$$

Multiply 1 times 3.

$$2x^{1}+3x^{0}+33x^{3-1}$$

Subtract 1 from 3.

$$2x^{1}+3x^{0}+33x^{2}$$

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

$$2x+3x^{0}+33x^{2}$$

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

$$2x+3\times 1+33x^{2}$$

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

$$2x+3+33x^{2}$$