Solve x^2(-2x)-2x(-2x)+3

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

Evaluate

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

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

Multiply x and x to get x^{2}.

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

Multiply 2 and -2 to get -4.

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

The opposite of -4x^{2} is 4x^{2}.

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

Differentiate w.r.t. x

$$2x\left(4-3x\right)$$

Steps Using Derivative Rule for Sum

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}.

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

Multiply 2 times -2x.

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

Subtract 1 from 2.

$$\left(-4x\right)x^{1}+4xx^{1-1}$$

Subtract 1 from 1.

$$\left(-4x\right)x^{1}+4xx^{0}$$

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

$$\left(-4x\right)x+4xx^{0}$$

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

$$\left(-4x\right)x+4x\times 1$$

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

$$\left(-4x\right)x+4x$$