Solve (3x^3+6x^2-7x-3)+(-7x^3-4x^2+9x-4)

$$( 3 x ^ { 3 } + 6 x ^ { 2 } - 7 x - 3 ) + ( - 7 x ^ { 3 } - 4 x ^ { 2 } + 9 x - 4 )$$

Evaluate

$$-4x^{3}+2x^{2}+2x-7$$

Solution Steps

Combine 3x^{3} and -7x^{3} to get -4x^{3}.

$$-4x^{3}+6x^{2}-7x-3-4x^{2}+9x-4$$

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

$$-4x^{3}+2x^{2}-7x-3+9x-4$$

Combine -7x and 9x to get 2x.

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

Subtract 4 from -3 to get -7.

$$-4x^{3}+2x^{2}+2x-7$$

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Sum

Combine 3x^{3} and -7x^{3} to get -4x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-4x^{3}+6x^{2}-7x-3-4x^{2}+9x-4)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(-4x^{3}+2x^{2}-7x-3+9x-4)$$

Combine -7x and 9x to get 2x.

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

Subtract 4 from -3 to get -7.

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

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}.

$$3\left(-4\right)x^{3-1}+2\times 2x^{2-1}+2x^{1-1}$$

Multiply 3 times -4.

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

Subtract 1 from 3.

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

Multiply 2 times 2.

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

Subtract 1 from 2.

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

Subtract 1 from 1.

$$-12x^{2}+4x^{1}+2x^{0}$$

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

$$-12x^{2}+4x+2x^{0}$$

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

$$-12x^{2}+4x+2\times 1$$

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

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