Solve (3x^3+11x^2-x-3)+(x^2+4x+1)

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

Evaluate

$$3x^{3}+12x^{2}+3x-2$$

Solution Steps

Combine 11x^{2} and x^{2} to get 12x^{2}.

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

Combine -x and 4x to get 3x.

$$3x^{3}+12x^{2}+3x-3+1$$

Add -3 and 1 to get -2.

$$3x^{3}+12x^{2}+3x-2$$

Differentiate w.r.t. x

$$9x^{2}+24x+3$$

Steps Using Derivative Rule for Sum

Combine 11x^{2} and x^{2} to get 12x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{3}+12x^{2}-x-3+4x+1)$$

Combine -x and 4x to get 3x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{3}+12x^{2}+3x-3+1)$$

Add -3 and 1 to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{3}+12x^{2}+3x-2)$$

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}.

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

Multiply 3 times 3.

$$9x^{3-1}+2\times 12x^{2-1}+3x^{1-1}$$

Subtract 1 from 3.

$$9x^{2}+2\times 12x^{2-1}+3x^{1-1}$$

Multiply 2 times 12.

$$9x^{2}+24x^{2-1}+3x^{1-1}$$

Subtract 1 from 2.

$$9x^{2}+24x^{1}+3x^{1-1}$$

Subtract 1 from 1.

$$9x^{2}+24x^{1}+3x^{0}$$

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

$$9x^{2}+24x+3x^{0}$$

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

$$9x^{2}+24x+3\times 1$$

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

$$9x^{2}+24x+3$$