Solve 2x^4-x^3+9x^2quadx^2+4

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

Evaluate

$$11x^{4}-x^{3}+4$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

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

Combine 2x^{4} and 9x^{4} to get 11x^{4}.

$$11x^{4}-x^{3}+4$$

Differentiate w.r.t. x

$$\left(44x-3\right)x^{2}$$

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}.

$$4\times 2x^{4-1}+3\left(-1\right)x^{3-1}+2\times 9x^{2}x^{2-1}$$

Multiply 4 times 2.

$$8x^{4-1}+3\left(-1\right)x^{3-1}+2\times 9x^{2}x^{2-1}$$

Subtract 1 from 4.

$$8x^{3}+3\left(-1\right)x^{3-1}+2\times 9x^{2}x^{2-1}$$

Multiply 3 times -1.

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

Subtract 1 from 3.

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

Multiply 3 times -1.

$$8x^{3}-3x^{2}+18x^{2}x^{2-1}$$

Subtract 1 from 2.

$$8x^{3}-3x^{2}+18x^{2}x^{1}$$

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

$$8x^{3}-3x^{2}+18x^{2}x$$