Solve (4x^3-5x^2+6x-3)+(7x^3+4x^2-7x+8)

$$( 4 x ^ { 3 } - 5 x ^ { 2 } + 6 x - 3 ) + ( 7 x ^ { 3 } + 4 x ^ { 2 } - 7 x + 8 )$$

Evaluate

$$11x^{3}-x^{2}-x+5$$

Solution Steps

Combine 4x^{3} and 7x^{3} to get 11x^{3}.

$$11x^{3}-5x^{2}+6x-3+4x^{2}-7x+8$$

Combine -5x^{2} and 4x^{2} to get -x^{2}.

$$11x^{3}-x^{2}+6x-3-7x+8$$

Combine 6x and -7x to get -x.

$$11x^{3}-x^{2}-x-3+8$$

Add -3 and 8 to get 5.

$$11x^{3}-x^{2}-x+5$$

Differentiate w.r.t. x

$$33x^{2}-2x-1$$

Steps Using Derivative Rule for Sum

Combine 4x^{3} and 7x^{3} to get 11x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(11x^{3}-5x^{2}+6x-3+4x^{2}-7x+8)$$

Combine -5x^{2} and 4x^{2} to get -x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(11x^{3}-x^{2}+6x-3-7x+8)$$

Combine 6x and -7x to get -x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(11x^{3}-x^{2}-x-3+8)$$

Add -3 and 8 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(11x^{3}-x^{2}-x+5)$$

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}.

$$3\times 11x^{3-1}+2\left(-1\right)x^{2-1}-x^{1-1}$$

Multiply 3 times 11.

$$33x^{3-1}+2\left(-1\right)x^{2-1}-x^{1-1}$$

Subtract 1 from 3.

$$33x^{2}+2\left(-1\right)x^{2-1}-x^{1-1}$$

Multiply 2 times -1.

$$33x^{2}-2x^{2-1}-x^{1-1}$$

Subtract 1 from 2.

$$33x^{2}-2x^{1}-x^{1-1}$$

Subtract 1 from 1.

$$33x^{2}-2x^{1}-x^{0}$$

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

$$33x^{2}-2x-x^{0}$$

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

$$33x^{2}-2x-1$$