Solve x^4+3x^3-x-3x^3+x^2-5x+3

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

Evaluate

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

Solution Steps

Combine 3x^{3} and -3x^{3} to get 0.

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

Combine -x and -5x to get -6x.

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

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Sum

Combine 3x^{3} and -3x^{3} to get 0.

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

Combine -x and -5x to get -6x.

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

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}.

$$4x^{4-1}-6x^{1-1}+2x^{2-1}$$

Subtract 1 from 4.

$$4x^{3}-6x^{1-1}+2x^{2-1}$$

Subtract 1 from 1.

$$4x^{3}-6x^{0}+2x^{2-1}$$

Subtract 1 from 2.

$$4x^{3}-6x^{0}+2x^{1}$$

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

$$4x^{3}-6x^{0}+2x$$

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

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