Solve 2x^3-7x^2-6+3x^3+4x^2-3x+3

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

Evaluate

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

Solution Steps

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

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

Combine -7x^{2} and 4x^{2} to get -3x^{2}.

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

Add -6 and 3 to get -3.

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

Differentiate w.r.t. x

$$15x^{2}-6x-3$$

Steps Using Derivative Rule for Sum

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

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

Combine -7x^{2} and 4x^{2} to get -3x^{2}.

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

Add -6 and 3 to get -3.

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

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}.

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

Multiply 3 times 5.

$$15x^{3-1}+2\left(-3\right)x^{2-1}-3x^{1-1}$$

Subtract 1 from 3.

$$15x^{2}+2\left(-3\right)x^{2-1}-3x^{1-1}$$

Multiply 2 times -3.

$$15x^{2}-6x^{2-1}-3x^{1-1}$$

Subtract 1 from 2.

$$15x^{2}-6x^{1}-3x^{1-1}$$

Subtract 1 from 1.

$$15x^{2}-6x^{1}-3x^{0}$$

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

$$15x^{2}-6x-3x^{0}$$

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

$$15x^{2}-6x-3$$