Solve (x^3+4x^2-10x+7)+(-6x^3+5x-10)

$$( { x }^{ 3 } +4 { x }^{ 2 } -10x+7)+(-6 { x }^{ 3 } +5x-10)$$

Evaluate

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

Solution Steps

Combine x^{3} and -6x^{3} to get -5x^{3}.

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

Combine -10x and 5x to get -5x.

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

Subtract 10 from 7 to get -3.

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

Differentiate w.r.t. x

$$-15x^{2}+8x-5$$

Steps Using Derivative Rule for Sum

Combine x^{3} and -6x^{3} to get -5x^{3}.

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

Combine -10x and 5x to get -5x.

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

Subtract 10 from 7 to get -3.

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

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}.

$$3\left(-5\right)x^{3-1}+2\times 4x^{2-1}-5x^{1-1}$$

Multiply 3 times -5.

$$-15x^{3-1}+2\times 4x^{2-1}-5x^{1-1}$$

Subtract 1 from 3.

$$-15x^{2}+2\times 4x^{2-1}-5x^{1-1}$$

Multiply 2 times 4.

$$-15x^{2}+8x^{2-1}-5x^{1-1}$$

Subtract 1 from 2.

$$-15x^{2}+8x^{1}-5x^{1-1}$$

Subtract 1 from 1.

$$-15x^{2}+8x^{1}-5x^{0}$$

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

$$-15x^{2}+8x-5x^{0}$$

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

$$-15x^{2}+8x-5$$