Solve (5x^3+7x-8)+(2x^3-5x^2-1x+3)

$$(5 { x }^{ 3 } +7x-8)+(2 { x }^{ 3 } -5 { x }^{ 2 } -1x+3)$$

Evaluate

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

Solution Steps

Combine 5x^{3} and 2x^{3} to get 7x^{3}.

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

Combine 7x and -x to get 6x.

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

Add -8 and 3 to get -5.

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

Differentiate w.r.t. x

$$21x^{2}-10x+6$$

Steps Using Derivative Rule for Sum

Combine 5x^{3} and 2x^{3} to get 7x^{3}.

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

Combine 7x and -x to get 6x.

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

Add -8 and 3 to get -5.

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

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}.

$$3\times 7x^{3-1}+6x^{1-1}+2\left(-5\right)x^{2-1}$$

Multiply 3 times 7.

$$21x^{3-1}+6x^{1-1}+2\left(-5\right)x^{2-1}$$

Subtract 1 from 3.

$$21x^{2}+6x^{1-1}+2\left(-5\right)x^{2-1}$$

Subtract 1 from 1.

$$21x^{2}+6x^{0}+2\left(-5\right)x^{2-1}$$

Multiply 1 times 6.

$$21x^{2}+6x^{0}-10x^{2-1}$$

Subtract 1 from 2.

$$21x^{2}+6x^{0}-10x^{1}$$

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

$$21x^{2}+6x^{0}-10x$$

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

$$21x^{2}+6\times 1-10x$$

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

$$21x^{2}+6-10x$$