Solve -3x^{4}+x^{3}-5x^2+12rightarrow-Q(x)=3x^4-x^3+5x^2-12

$$- 3 x ^ { 4 } + x ^ { 3 } - 5 x ^ { 2 } + 12 \rightarrow - Q ( x ) = 3 x ^ { 4 } - x ^ { 3 } + 5 x ^ { 2 } - 12$$

Differentiate w.r.t. x

$$x\left(-12x^{2}+3x-10\right)$$

Steps Using Derivative Rule for Sum

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}.

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

Multiply 4 times -3.

$$-12x^{4-1}+3x^{3-1}+2\left(-5\right)x^{2-1}$$

Subtract 1 from 4.

$$-12x^{3}+3x^{3-1}+2\left(-5\right)x^{2-1}$$

Subtract 1 from 3.

$$-12x^{3}+3x^{2}+2\left(-5\right)x^{2-1}$$

Multiply 3 times 1.

$$-12x^{3}+3x^{2}-10x^{2-1}$$

Subtract 1 from 2.

$$-12x^{3}+3x^{2}-10x^{1}$$

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

$$-12x^{3}+3x^{2}-10x$$

Evaluate

$$12-5x^{2}+x^{3}-3x^{4}$$