Solve (4x^4-5x^3+3x+12)+(19x^4-2x^3-5x-4)

$$( 4 x ^ { 4 } - 5 x ^ { 3 } + 3 x + 12 ) + ( 19 x ^ { 4 } - 2 x ^ { 3 } - 5 x - 4 )$$

Evaluate

$$23x^{4}-7x^{3}-2x+8$$

Solution Steps

Combine 4x^{4} and 19x^{4} to get 23x^{4}.

$$23x^{4}-5x^{3}+3x+12-2x^{3}-5x-4$$

Combine -5x^{3} and -2x^{3} to get -7x^{3}.

$$23x^{4}-7x^{3}+3x+12-5x-4$$

Combine 3x and -5x to get -2x.

$$23x^{4}-7x^{3}-2x+12-4$$

Subtract 4 from 12 to get 8.

$$23x^{4}-7x^{3}-2x+8$$

Differentiate w.r.t. x

$$92x^{3}-21x^{2}-2$$

Steps Using Derivative Rule for Sum

Combine 4x^{4} and 19x^{4} to get 23x^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(23x^{4}-5x^{3}+3x+12-2x^{3}-5x-4)$$

Combine -5x^{3} and -2x^{3} to get -7x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(23x^{4}-7x^{3}+3x+12-5x-4)$$

Combine 3x and -5x to get -2x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(23x^{4}-7x^{3}-2x+12-4)$$

Subtract 4 from 12 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(23x^{4}-7x^{3}-2x+8)$$

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}.

$$4\times 23x^{4-1}+3\left(-7\right)x^{3-1}-2x^{1-1}$$

Multiply 4 times 23.

$$92x^{4-1}+3\left(-7\right)x^{3-1}-2x^{1-1}$$

Subtract 1 from 4.

$$92x^{3}+3\left(-7\right)x^{3-1}-2x^{1-1}$$

Multiply 3 times -7.

$$92x^{3}-21x^{3-1}-2x^{1-1}$$

Subtract 1 from 3.

$$92x^{3}-21x^{2}-2x^{1-1}$$

Subtract 1 from 1.

$$92x^{3}-21x^{2}-2x^{0}$$

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

$$92x^{3}-21x^{2}-2$$