Solve (4x^3-2x^2+3x-1)+(5x^2-x-5)

$$( 4 x ^ { 3 } - 2 x ^ { 2 } + 3 x - 1 ) + ( 5 x ^ { 2 } - x - 5 )$$

Evaluate

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

Solution Steps

Combine -2x^{2} and 5x^{2} to get 3x^{2}.

$$4x^{3}+3x^{2}+3x-1-x-5$$

Combine 3x and -x to get 2x.

$$4x^{3}+3x^{2}+2x-1-5$$

Subtract 5 from -1 to get -6.

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

Differentiate w.r.t. x

$$12x^{2}+6x+2$$

Steps Using Derivative Rule for Sum

Combine -2x^{2} and 5x^{2} to get 3x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{3}+3x^{2}+3x-1-x-5)$$

Combine 3x and -x to get 2x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{3}+3x^{2}+2x-1-5)$$

Subtract 5 from -1 to get -6.

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

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}.

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

Multiply 3 times 4.

$$12x^{3-1}+2\times 3x^{2-1}+2x^{1-1}$$

Subtract 1 from 3.

$$12x^{2}+2\times 3x^{2-1}+2x^{1-1}$$

Multiply 2 times 3.

$$12x^{2}+6x^{2-1}+2x^{1-1}$$

Subtract 1 from 2.

$$12x^{2}+6x^{1}+2x^{1-1}$$

Subtract 1 from 1.

$$12x^{2}+6x^{1}+2x^{0}$$

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

$$12x^{2}+6x+2x^{0}$$

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

$$12x^{2}+6x+2\times 1$$

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

$$12x^{2}+6x+2$$