Solve 2x^{2}-x^3+3x^3-x-x^3+4x+3-5x^2=

$$2 x ^ { 2 } - x ^ { 3 } + 3 x ^ { 3 } - x - x ^ { 3 } + 4 x + 3 - 5 x ^ { 2 } =$$

Evaluate

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

Solution Steps

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

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

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

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

Combine -x and 4x to get 3x.

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

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

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

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Sum

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

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

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

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

Combine -x and 4x to get 3x.

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

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

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

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}.

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

Multiply 2 times -3.

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

Subtract 1 from 2.

$$-6x^{1}+3x^{3-1}+3x^{1-1}$$

Subtract 1 from 3.

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

Subtract 1 from 1.

$$-6x^{1}+3x^{2}+3x^{0}$$

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

$$-6x+3x^{2}+3x^{0}$$

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

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

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

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