Solve 2x^3+3x^2-5x+x^3-7x^2+8

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

Evaluate

$$3x^{3}-4x^{2}-5x+8$$

Solution Steps

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

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

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

$$3x^{3}-4x^{2}-5x+8$$

Differentiate w.r.t. x

$$9x^{2}-8x-5$$

Steps Using Derivative Rule for Sum

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

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

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{3}-4x^{2}-5x+8)$$

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}.

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

Multiply 3 times 3.

$$9x^{3-1}+2\left(-4\right)x^{2-1}-5x^{1-1}$$

Subtract 1 from 3.

$$9x^{2}+2\left(-4\right)x^{2-1}-5x^{1-1}$$

Multiply 2 times -4.

$$9x^{2}-8x^{2-1}-5x^{1-1}$$

Subtract 1 from 2.

$$9x^{2}-8x^{1}-5x^{1-1}$$

Subtract 1 from 1.

$$9x^{2}-8x^{1}-5x^{0}$$

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

$$9x^{2}-8x-5x^{0}$$

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

$$9x^{2}-8x-5$$