Solve (6x^3-4x^2+x-9)+(3x^2+7x+3)

$$(6 { x }^{ 3 } -4 { x }^{ 2 } +x-9)+(3 { x }^{ 2 } +7x+3)$$

Evaluate

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

Solution Steps

Combine -4x^{2} and 3x^{2} to get -x^{2}.

$$6x^{3}-x^{2}+x-9+7x+3$$

Combine x and 7x to get 8x.

$$6x^{3}-x^{2}+8x-9+3$$

Add -9 and 3 to get -6.

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

Differentiate w.r.t. x

$$18x^{2}-2x+8$$

Steps Using Derivative Rule for Sum

Combine -4x^{2} and 3x^{2} to get -x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{3}-x^{2}+x-9+7x+3)$$

Combine x and 7x to get 8x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{3}-x^{2}+8x-9+3)$$

Add -9 and 3 to get -6.

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

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}.

$$3\times 6x^{3-1}+2\left(-1\right)x^{2-1}+8x^{1-1}$$

Multiply 3 times 6.

$$18x^{3-1}+2\left(-1\right)x^{2-1}+8x^{1-1}$$

Subtract 1 from 3.

$$18x^{2}+2\left(-1\right)x^{2-1}+8x^{1-1}$$

Multiply 2 times -1.

$$18x^{2}-2x^{2-1}+8x^{1-1}$$

Subtract 1 from 2.

$$18x^{2}-2x^{1}+8x^{1-1}$$

Subtract 1 from 1.

$$18x^{2}-2x^{1}+8x^{0}$$

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

$$18x^{2}-2x+8x^{0}$$

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

$$18x^{2}-2x+8\times 1$$

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

$$18x^{2}-2x+8$$