Solve x^3-3x^2-x+4-4x+5

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

Evaluate

$$x^{3}-3x^{2}-5x+9$$

Solution Steps

Combine -x and -4x to get -5x.

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

Add 4 and 5 to get 9.

$$x^{3}-3x^{2}-5x+9$$

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Sum

Combine -x and -4x to get -5x.

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

Add 4 and 5 to get 9.

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

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}.

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

Subtract 1 from 3.

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

Multiply 2 times -3.

$$3x^{2}-6x^{2-1}-5x^{1-1}$$

Subtract 1 from 2.

$$3x^{2}-6x^{1}-5x^{1-1}$$

Subtract 1 from 1.

$$3x^{2}-6x^{1}-5x^{0}$$

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

$$3x^{2}-6x-5x^{0}$$

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

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