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

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

Evaluate

$$x-2$$

Solution Steps

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 1 to the power of 2 and get 1.

$$x-1-1$$

Subtract 1 from -1 to get -2.

$$x-2$$

Differentiate w.r.t. x

$$1$$

Steps Using Derivative Rule for Sum

Consider \left(\sqrt{x-1}-1\right)\left(\sqrt{x-1}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 1 to the power of 2 and get 1.

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

Subtract 1 from -1 to get -2.

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

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}.

$$x^{1-1}$$

Subtract 1 from 1.

$$x^{0}$$

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

$$1$$