Solve 7x-3x^2-8+5x^3+(2x+x^2)-4x^3-1

$$7 x - 3 x ^ { 2 } - 8 + 5 x ^ { 3 } + ( 2 x + x ^ { 2 } ) - 4 x ^ { 3 } - 1$$

Evaluate

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

Solution Steps

Combine 7x and 2x to get 9x.

$$9x-3x^{2}-8+5x^{3}+x^{2}-4x^{3}-1$$

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

$$9x-2x^{2}-8+5x^{3}-4x^{3}-1$$

Combine 5x^{3} and -4x^{3} to get x^{3}.

$$9x-2x^{2}-8+x^{3}-1$$

Subtract 1 from -8 to get -9.

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

Differentiate w.r.t. x

$$3x^{2}-4x+9$$

Steps Using Derivative Rule for Sum

Combine 7x and 2x to get 9x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(9x-3x^{2}-8+5x^{3}+x^{2}-4x^{3}-1)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(9x-2x^{2}-8+5x^{3}-4x^{3}-1)$$

Combine 5x^{3} and -4x^{3} to get x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(9x-2x^{2}-8+x^{3}-1)$$

Subtract 1 from -8 to get -9.

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

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}.

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

Subtract 1 from 1.

$$9x^{0}+2\left(-2\right)x^{2-1}+3x^{3-1}$$

Multiply 2 times -2.

$$9x^{0}-4x^{2-1}+3x^{3-1}$$

Subtract 1 from 2.

$$9x^{0}-4x^{1}+3x^{3-1}$$

Subtract 1 from 3.

$$9x^{0}-4x^{1}+3x^{2}$$

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

$$9x^{0}-4x+3x^{2}$$

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

$$9\times 1-4x+3x^{2}$$

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

$$9-4x+3x^{2}$$