Solve (4x^3-2x^2+5x-1)+(2x^3+7x^2-x+4)

$$( 4 x ^ { 3 } - 2 x ^ { 2 } + 5 x - 1 ) + ( 2 x ^ { 3 } + 7 x ^ { 2 } - x + 4 )$$

Evaluate

$$6x^{3}+5x^{2}+4x+3$$

Solution Steps

Combine 4x^{3} and 2x^{3} to get 6x^{3}.

$$6x^{3}-2x^{2}+5x-1+7x^{2}-x+4$$

Combine -2x^{2} and 7x^{2} to get 5x^{2}.

$$6x^{3}+5x^{2}+5x-1-x+4$$

Combine 5x and -x to get 4x.

$$6x^{3}+5x^{2}+4x-1+4$$

Add -1 and 4 to get 3.

$$6x^{3}+5x^{2}+4x+3$$

Differentiate w.r.t. x

$$18x^{2}+10x+4$$

Steps Using Derivative Rule for Sum

Combine 4x^{3} and 2x^{3} to get 6x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{3}-2x^{2}+5x-1+7x^{2}-x+4)$$

Combine -2x^{2} and 7x^{2} to get 5x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{3}+5x^{2}+5x-1-x+4)$$

Combine 5x and -x to get 4x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{3}+5x^{2}+4x-1+4)$$

Add -1 and 4 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{3}+5x^{2}+4x+3)$$

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}.

$$3\times 6x^{3-1}+2\times 5x^{2-1}+4x^{1-1}$$

Multiply 3 times 6.

$$18x^{3-1}+2\times 5x^{2-1}+4x^{1-1}$$

Subtract 1 from 3.

$$18x^{2}+2\times 5x^{2-1}+4x^{1-1}$$

Multiply 2 times 5.

$$18x^{2}+10x^{2-1}+4x^{1-1}$$

Subtract 1 from 2.

$$18x^{2}+10x^{1}+4x^{1-1}$$

Subtract 1 from 1.

$$18x^{2}+10x^{1}+4x^{0}$$

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

$$18x^{2}+10x+4x^{0}$$

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

$$18x^{2}+10x+4\times 1$$

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

$$18x^{2}+10x+4$$