Solve (8x^3-7x^2+6x+2)+(-9x^3-3x^2+7x-5)

$$( 8 x ^ { 3 } - 7 x ^ { 2 } + 6 x + 2 ) + ( - 9 x ^ { 3 } - 3 x ^ { 2 } + 7 x - 5 )$$

Evaluate

$$-x^{3}-10x^{2}+13x-3$$

Solution Steps

Combine 8x^{3} and -9x^{3} to get -x^{3}.

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

Combine -7x^{2} and -3x^{2} to get -10x^{2}.

$$-x^{3}-10x^{2}+6x+2+7x-5$$

Combine 6x and 7x to get 13x.

$$-x^{3}-10x^{2}+13x+2-5$$

Subtract 5 from 2 to get -3.

$$-x^{3}-10x^{2}+13x-3$$

Differentiate w.r.t. x

$$13-20x-3x^{2}$$

Steps Using Derivative Rule for Sum

Combine 8x^{3} and -9x^{3} to get -x^{3}.

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

Combine -7x^{2} and -3x^{2} to get -10x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-x^{3}-10x^{2}+6x+2+7x-5)$$

Combine 6x and 7x to get 13x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-x^{3}-10x^{2}+13x+2-5)$$

Subtract 5 from 2 to get -3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-x^{3}-10x^{2}+13x-3)$$

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}.

$$3\left(-1\right)x^{3-1}+2\left(-10\right)x^{2-1}+13x^{1-1}$$

Multiply 3 times -1.

$$-3x^{3-1}+2\left(-10\right)x^{2-1}+13x^{1-1}$$

Subtract 1 from 3.

$$-3x^{2}+2\left(-10\right)x^{2-1}+13x^{1-1}$$

Multiply 2 times -10.

$$-3x^{2}-20x^{2-1}+13x^{1-1}$$

Subtract 1 from 2.

$$-3x^{2}-20x^{1}+13x^{1-1}$$

Subtract 1 from 1.

$$-3x^{2}-20x^{1}+13x^{0}$$

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

$$-3x^{2}-20x+13x^{0}$$

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

$$-3x^{2}-20x+13\times 1$$

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

$$-3x^{2}-20x+13$$