Solve 6x^3+2x^2+18x+3div3x+1

$$6 { x }^{ 3 } +2 { x }^{ 2 } +18x+3 \div 3x+1$$

Evaluate

$$6x^{3}+2x^{2}+19x+1$$

Solution Steps

Divide 3 by 3 to get 1.

$$6x^{3}+2x^{2}+18x+1x+1$$

Combine 18x and 1x to get 19x.

$$6x^{3}+2x^{2}+19x+1$$

Differentiate w.r.t. x

$$18x^{2}+4x+19$$

Steps Using Derivative Rule for Sum

Divide 3 by 3 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{3}+2x^{2}+18x+1x+1)$$

Combine 18x and 1x to get 19x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{3}+2x^{2}+19x+1)$$

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}.

$$3\times 6x^{3-1}+2\times 2x^{2-1}+19x^{1-1}$$

Multiply 3 times 6.

$$18x^{3-1}+2\times 2x^{2-1}+19x^{1-1}$$

Subtract 1 from 3.

$$18x^{2}+2\times 2x^{2-1}+19x^{1-1}$$

Multiply 2 times 2.

$$18x^{2}+4x^{2-1}+19x^{1-1}$$

Subtract 1 from 2.

$$18x^{2}+4x^{1}+19x^{1-1}$$

Subtract 1 from 1.

$$18x^{2}+4x^{1}+19x^{0}$$

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

$$18x^{2}+4x+19x^{0}$$

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

$$18x^{2}+4x+19\times 1$$

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

$$18x^{2}+4x+19$$