Solve (x-x^{2}+x^4)+(x^3-7x^2+7)+(8x^2-4x+8)

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

Evaluate

$$x^{4}+x^{3}-3x+15$$

Solution Steps

Combine -x^{2} and -7x^{2} to get -8x^{2}.

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

Combine -8x^{2} and 8x^{2} to get 0.

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

Combine x and -4x to get -3x.

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

Add 7 and 8 to get 15.

$$-3x+x^{4}+x^{3}+15$$

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Sum

Combine -x^{2} and -7x^{2} to get -8x^{2}.

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

Combine -8x^{2} and 8x^{2} to get 0.

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

Combine x and -4x to get -3x.

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

Add 7 and 8 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-3x+x^{4}+x^{3}+15)$$

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}.

$$-3x^{1-1}+4x^{4-1}+3x^{3-1}$$

Subtract 1 from 1.

$$-3x^{0}+4x^{4-1}+3x^{3-1}$$

Subtract 1 from 4.

$$-3x^{0}+4x^{3}+3x^{3-1}$$

Subtract 1 from 3.

$$-3x^{0}+4x^{3}+3x^{2}$$

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

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