Solve 3x^2-2x+3-4x^3+4x+x^2-7+3x^3

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

Evaluate

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

Solution Steps

Combine -2x and 4x to get 2x.

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

Combine 3x^{2} and x^{2} to get 4x^{2}.

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

Subtract 7 from 3 to get -4.

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

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

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

Differentiate w.r.t. x

$$2+8x-3x^{2}$$

Steps Using Derivative Rule for Sum

Combine -2x and 4x to get 2x.

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

Combine 3x^{2} and x^{2} to get 4x^{2}.

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

Subtract 7 from 3 to get -4.

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

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

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

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}.

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

Multiply 2 times 4.

$$8x^{2-1}+2x^{1-1}+3\left(-1\right)x^{3-1}$$

Subtract 1 from 2.

$$8x^{1}+2x^{1-1}+3\left(-1\right)x^{3-1}$$

Subtract 1 from 1.

$$8x^{1}+2x^{0}+3\left(-1\right)x^{3-1}$$

Multiply 1 times 2.

$$8x^{1}+2x^{0}-3x^{3-1}$$

Subtract 1 from 3.

$$8x^{1}+2x^{0}-3x^{2}$$

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

$$8x+2x^{0}-3x^{2}$$

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

$$8x+2\times 1-3x^{2}$$

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

$$8x+2-3x^{2}$$