Solve x^4-3x^2+5-x^3-2x^4+6{x}^2+{x}^3+8x-6=

$${ x }^{ 4 } -3 { x }^{ 2 } +5- { x }^{ 3 } -2 { x }^{ 4 } +6 { x }^{ 2 } + { x }^{ 3 } +8x-6=$$

Evaluate

$$-x^{4}+3x^{2}+8x-1$$

Solution Steps

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

$$-x^{4}-3x^{2}+5-x^{3}+6x^{2}+x^{3}+8x-6$$

Combine -3x^{2} and 6x^{2} to get 3x^{2}.

$$-x^{4}+3x^{2}+5-x^{3}+x^{3}+8x-6$$

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

$$-x^{4}+3x^{2}+5+8x-6$$

Subtract 6 from 5 to get -1.

$$-x^{4}+3x^{2}-1+8x$$

Differentiate w.r.t. x

$$8+6x-4x^{3}$$

Steps Using Derivative Rule for Sum

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(-x^{4}-3x^{2}+5-x^{3}+6x^{2}+x^{3}+8x-6)$$

Combine -3x^{2} and 6x^{2} to get 3x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-x^{4}+3x^{2}+5-x^{3}+x^{3}+8x-6)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(-x^{4}+3x^{2}+5+8x-6)$$

Subtract 6 from 5 to get -1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-x^{4}+3x^{2}-1+8x)$$

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}.

$$4\left(-1\right)x^{4-1}+2\times 3x^{2-1}+8x^{1-1}$$

Multiply 4 times -1.

$$-4x^{4-1}+2\times 3x^{2-1}+8x^{1-1}$$

Subtract 1 from 4.

$$-4x^{3}+2\times 3x^{2-1}+8x^{1-1}$$

Multiply 2 times 3.

$$-4x^{3}+6x^{2-1}+8x^{1-1}$$

Subtract 1 from 2.

$$-4x^{3}+6x^{1}+8x^{1-1}$$

Subtract 1 from 1.

$$-4x^{3}+6x^{1}+8x^{0}$$

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

$$-4x^{3}+6x+8x^{0}$$

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

$$-4x^{3}+6x+8\times 1$$

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

$$-4x^{3}+6x+8$$