Solve (6x^4+2x^3-x+5)+(-2x^4+x^3+4x-10)

$$( 6 x ^ { 4 } + 2 x ^ { 3 } - x + 5 ) + ( - 2 x ^ { 4 } + x ^ { 3 } + 4 x - 10 )$$

Evaluate

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

Solution Steps

Combine 6x^{4} and -2x^{4} to get 4x^{4}.

$$4x^{4}+2x^{3}-x+5+x^{3}+4x-10$$

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

$$4x^{4}+3x^{3}-x+5+4x-10$$

Combine -x and 4x to get 3x.

$$4x^{4}+3x^{3}+3x+5-10$$

Subtract 10 from 5 to get -5.

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

Differentiate w.r.t. x

$$16x^{3}+9x^{2}+3$$

Steps Using Derivative Rule for Sum

Combine 6x^{4} and -2x^{4} to get 4x^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{4}+2x^{3}-x+5+x^{3}+4x-10)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{4}+3x^{3}-x+5+4x-10)$$

Combine -x and 4x to get 3x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{4}+3x^{3}+3x+5-10)$$

Subtract 10 from 5 to get -5.

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

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}.

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

Multiply 4 times 4.

$$16x^{4-1}+3\times 3x^{3-1}+3x^{1-1}$$

Subtract 1 from 4.

$$16x^{3}+3\times 3x^{3-1}+3x^{1-1}$$

Multiply 3 times 3.

$$16x^{3}+9x^{3-1}+3x^{1-1}$$

Subtract 1 from 3.

$$16x^{3}+9x^{2}+3x^{1-1}$$

Subtract 1 from 1.

$$16x^{3}+9x^{2}+3x^{0}$$

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

$$16x^{3}+9x^{2}+3\times 1$$

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

$$16x^{3}+9x^{2}+3$$