Solve (2x^3-x^2+3x-6)+(4x^2+x+4)

$$( 2 x ^ { 3 } - x ^ { 2 } + 3 x - 6 ) + ( 4 x ^ { 2 } + x + 4 )$$

Evaluate

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

Solution Steps

Combine -x^{2} and 4x^{2} to get 3x^{2}.

$$2x^{3}+3x^{2}+3x-6+x+4$$

Combine 3x and x to get 4x.

$$2x^{3}+3x^{2}+4x-6+4$$

Add -6 and 4 to get -2.

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

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Sum

Combine -x^{2} and 4x^{2} to get 3x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{3}+3x^{2}+3x-6+x+4)$$

Combine 3x and x to get 4x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{3}+3x^{2}+4x-6+4)$$

Add -6 and 4 to get -2.

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

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}.

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

Multiply 3 times 2.

$$6x^{3-1}+2\times 3x^{2-1}+4x^{1-1}$$

Subtract 1 from 3.

$$6x^{2}+2\times 3x^{2-1}+4x^{1-1}$$

Multiply 2 times 3.

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

Subtract 1 from 2.

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

Subtract 1 from 1.

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

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

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

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

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

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

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