Solve =x^2+3x^3-12x^2+52x-48

$$= x ^ { 2 } + 3 x ^ { 3 } - 12 x ^ { 2 } + 52 x - 48$$

Differentiate w.r.t. x

$$9x^{2}-22x+52$$

Steps Using Derivative Rule for Sum

Combine x^{2} and -12x^{2} to get -11x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-11x^{2}+3x^{3}+52x-48)$$

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}.

$$2\left(-11\right)x^{2-1}+3\times 3x^{3-1}+52x^{1-1}$$

Multiply 2 times -11.

$$-22x^{2-1}+3\times 3x^{3-1}+52x^{1-1}$$

Subtract 1 from 2.

$$-22x^{1}+3\times 3x^{3-1}+52x^{1-1}$$

Multiply 3 times 3.

$$-22x^{1}+9x^{3-1}+52x^{1-1}$$

Subtract 1 from 3.

$$-22x^{1}+9x^{2}+52x^{1-1}$$

Subtract 1 from 1.

$$-22x^{1}+9x^{2}+52x^{0}$$

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

$$-22x+9x^{2}+52x^{0}$$

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

$$-22x+9x^{2}+52\times 1$$

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

$$-22x+9x^{2}+52$$

Evaluate

$$3x^{3}-11x^{2}+52x-48$$

Solution Steps

Combine x^{2} and -12x^{2} to get -11x^{2}.

$$-11x^{2}+3x^{3}+52x-48$$