Solve (5x^{4}+2x^3+3x^2+12)+(7x^3+4x^2+3)

$$( 5 x ^ { 4 } + 2 x ^ { 3 } + 3 x ^ { 2 } + 12 ) + ( 7 x ^ { 3 } + 4 x ^ { 2 } + 3 )$$

Evaluate

$$5x^{4}+9x^{3}+7x^{2}+15$$

Solution Steps

Combine 2x^{3} and 7x^{3} to get 9x^{3}.

$$5x^{4}+9x^{3}+3x^{2}+12+4x^{2}+3$$

Combine 3x^{2} and 4x^{2} to get 7x^{2}.

$$5x^{4}+9x^{3}+7x^{2}+12+3$$

Add 12 and 3 to get 15.

$$5x^{4}+9x^{3}+7x^{2}+15$$

Differentiate w.r.t. x

$$x\left(20x^{2}+27x+14\right)$$

Steps Using Derivative Rule for Sum

Combine 2x^{3} and 7x^{3} to get 9x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{4}+9x^{3}+3x^{2}+12+4x^{2}+3)$$

Combine 3x^{2} and 4x^{2} to get 7x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{4}+9x^{3}+7x^{2}+12+3)$$

Add 12 and 3 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{4}+9x^{3}+7x^{2}+15)$$

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}.

$$4\times 5x^{4-1}+3\times 9x^{3-1}+2\times 7x^{2-1}$$

Multiply 4 times 5.

$$20x^{4-1}+3\times 9x^{3-1}+2\times 7x^{2-1}$$

Subtract 1 from 4.

$$20x^{3}+3\times 9x^{3-1}+2\times 7x^{2-1}$$

Multiply 3 times 9.

$$20x^{3}+27x^{3-1}+2\times 7x^{2-1}$$

Subtract 1 from 3.

$$20x^{3}+27x^{2}+2\times 7x^{2-1}$$

Multiply 3 times 9.

$$20x^{3}+27x^{2}+14x^{2-1}$$

Subtract 1 from 2.

$$20x^{3}+27x^{2}+14x^{1}$$

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

$$20x^{3}+27x^{2}+14x$$