Solve (-sqrt{1+x}-x+1)*(sqrt{1+x}-x+1)

$$(- \sqrt{ 1+x } -x+1) \times ( \sqrt{ 1+x } -x+1)$$

Evaluate

$$x\left(x-3\right)$$

Solution Steps

Apply the distributive property by multiplying each term of -\sqrt{1+x}-x+1 by each term of \sqrt{1+x}-x+1.

$$\left(-\sqrt{1+x}\right)\sqrt{1+x}-\left(-\sqrt{1+x}\right)x-\sqrt{1+x}-x\sqrt{1+x}+x^{2}-x+\sqrt{1+x}-x+1$$

Multiply -1 and -1 to get 1.

$$\left(-\sqrt{1+x}\right)\sqrt{1+x}+\sqrt{1+x}x-\sqrt{1+x}-x\sqrt{1+x}+x^{2}-x+\sqrt{1+x}-x+1$$

Combine \sqrt{1+x}x and -x\sqrt{1+x} to get 0.

$$\left(-\sqrt{1+x}\right)\sqrt{1+x}-\sqrt{1+x}+x^{2}-x+\sqrt{1+x}-x+1$$

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

$$\left(-\sqrt{1+x}\right)\sqrt{1+x}-\sqrt{1+x}+x^{2}-2x+\sqrt{1+x}+1$$

Multiply \sqrt{1+x} and \sqrt{1+x} to get \left(\sqrt{1+x}\right)^{2}.

$$-\left(\sqrt{1+x}\right)^{2}-\sqrt{1+x}+x^{2}-2x+\sqrt{1+x}+1$$

Combine -\sqrt{1+x} and \sqrt{1+x} to get 0.

$$-\left(\sqrt{1+x}\right)^{2}+x^{2}-2x+1$$

Calculate \sqrt{1+x} to the power of 2 and get 1+x.

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

To find the opposite of 1+x, find the opposite of each term.

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

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

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

Add -1 and 1 to get 0.

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

Differentiate w.r.t. x

$$2x-3$$

Steps Using Derivative Rule for Sum

Apply the distributive property by multiplying each term of -\sqrt{1+x}-x+1 by each term of \sqrt{1+x}-x+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\sqrt{1+x}\right)\sqrt{1+x}-\left(-\sqrt{1+x}\right)x-\sqrt{1+x}-x\sqrt{1+x}+x^{2}-x+\sqrt{1+x}-x+1)$$

Multiply -1 and -1 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\sqrt{1+x}\right)\sqrt{1+x}+\sqrt{1+x}x-\sqrt{1+x}-x\sqrt{1+x}+x^{2}-x+\sqrt{1+x}-x+1)$$

Combine \sqrt{1+x}x and -x\sqrt{1+x} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\sqrt{1+x}\right)\sqrt{1+x}-\sqrt{1+x}+x^{2}-x+\sqrt{1+x}-x+1)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\sqrt{1+x}\right)\sqrt{1+x}-\sqrt{1+x}+x^{2}-2x+\sqrt{1+x}+1)$$

Multiply \sqrt{1+x} and \sqrt{1+x} to get \left(\sqrt{1+x}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-\left(\sqrt{1+x}\right)^{2}-\sqrt{1+x}+x^{2}-2x+\sqrt{1+x}+1)$$

Combine -\sqrt{1+x} and \sqrt{1+x} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-\left(\sqrt{1+x}\right)^{2}+x^{2}-2x+1)$$

Calculate \sqrt{1+x} to the power of 2 and get 1+x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-\left(1+x\right)+x^{2}-2x+1)$$

To find the opposite of 1+x, find the opposite of each term.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-1-x+x^{2}-2x+1)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(-1-3x+x^{2}+1)$$

Add -1 and 1 to get 0.

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

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}.

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

Subtract 1 from 1.

$$-3x^{0}+2x^{2-1}$$

Subtract 1 from 2.

$$-3x^{0}+2x^{1}$$

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

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

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

$$-3+2x$$