Solve x+4x^3-24x^2+12x-32

$$x + 4 x ^ { 3 } - 24 x ^ { 2 } + 12 x - 32$$

Differentiate w.r.t. x

$$12x^{2}-48x+13$$

Steps Using Derivative Rule for Sum

Combine x and 12x to get 13x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(13x+4x^{3}-24x^{2}-32)$$

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}.

$$13x^{1-1}+3\times 4x^{3-1}+2\left(-24\right)x^{2-1}$$

Subtract 1 from 1.

$$13x^{0}+3\times 4x^{3-1}+2\left(-24\right)x^{2-1}$$

Multiply 3 times 4.

$$13x^{0}+12x^{3-1}+2\left(-24\right)x^{2-1}$$

Subtract 1 from 3.

$$13x^{0}+12x^{2}+2\left(-24\right)x^{2-1}$$

Multiply 3 times 4.

$$13x^{0}+12x^{2}-48x^{2-1}$$

Subtract 1 from 2.

$$13x^{0}+12x^{2}-48x^{1}$$

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

$$13x^{0}+12x^{2}-48x$$

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

$$13\times 1+12x^{2}-48x$$

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

$$13+12x^{2}-48x$$

Evaluate

$$4x^{3}-24x^{2}+13x-32$$

Solution Steps

Combine x and 12x to get 13x.

$$13x+4x^{3}-24x^{2}-32$$