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

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

Evaluate

$$7x^{3}+3x^{2}-10x+1$$

Solution Steps

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

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

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

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

Combine -7x and -3x to get -10x.

$$7x^{3}+3x^{2}-10x+5-4$$

Subtract 4 from 5 to get 1.

$$7x^{3}+3x^{2}-10x+1$$

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Sum

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

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

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

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

Combine -7x and -3x to get -10x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(7x^{3}+3x^{2}-10x+5-4)$$

Subtract 4 from 5 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(7x^{3}+3x^{2}-10x+1)$$

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}.

$$3\times 7x^{3-1}+2\times 3x^{2-1}-10x^{1-1}$$

Multiply 3 times 7.

$$21x^{3-1}+2\times 3x^{2-1}-10x^{1-1}$$

Subtract 1 from 3.

$$21x^{2}+2\times 3x^{2-1}-10x^{1-1}$$

Multiply 2 times 3.

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

Subtract 1 from 2.

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

Subtract 1 from 1.

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

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

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

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

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