Solve 2-2x^2-x+x^3-3x^2-4x^3

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

Evaluate

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

Solution Steps

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

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

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

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

Differentiate w.r.t. x

$$\left(-x-1\right)\left(9x+1\right)$$

Steps Using Derivative Rule for Sum

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

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

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

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

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}.

$$2\left(-5\right)x^{2-1}-x^{1-1}+3\left(-3\right)x^{3-1}$$

Multiply 2 times -5.

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

Subtract 1 from 2.

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

Subtract 1 from 1.

$$-10x^{1}-x^{0}+3\left(-3\right)x^{3-1}$$

Multiply 1 times -1.

$$-10x^{1}-x^{0}-9x^{3-1}$$

Subtract 1 from 3.

$$-10x^{1}-x^{0}-9x^{2}$$

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

$$-10x-x^{0}-9x^{2}$$

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

$$-10x-1-9x^{2}$$