Solve {l}{-x^2+x+3}{--2x^2+4x+5}

$$\left. \begin{array} { l } { - x ^ { 2 } + x + 3 } \\ { - - 2 x ^ { 2 } + 4 x + 5 } \end{array} \right.$$

Evaluate

$$x^{2}+5x+8$$

Solution Steps

The opposite of -2x^{2} is 2x^{2}.

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

Combine x and 4x to get 5x.

$$-x^{2}+5x+3+2x^{2}+5$$

Add 3 and 5 to get 8.

$$-x^{2}+5x+8+2x^{2}$$

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

$$x^{2}+5x+8$$

Differentiate w.r.t. x

$$2x+5$$

Steps Using Derivative Rule for Sum

The opposite of -2x^{2} is 2x^{2}.

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

Combine x and 4x to get 5x.

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

Add 3 and 5 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-x^{2}+5x+8+2x^{2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+5x+8)$$

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}.

$$2x^{2-1}+5x^{1-1}$$

Subtract 1 from 2.

$$2x^{1}+5x^{1-1}$$

Subtract 1 from 1.

$$2x^{1}+5x^{0}$$

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

$$2x+5x^{0}$$

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

$$2x+5\times 1$$

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

$$2x+5$$