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

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

Evaluate

$$x-1$$

Solution Steps

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

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

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

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

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

$$-1+x$$

Differentiate w.r.t. x

$$1$$

Steps Using Derivative Rule for Sum

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

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

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

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

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

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

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}.

$$x^{1-1}$$

Subtract 1 from 1.

$$x^{0}$$

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

$$1$$