Solve (sqrt{x+2}-2)(sqrt{x+2}+2)

$$( \sqrt{ x+2 } -2)( \sqrt{ x+2 } +2)$$

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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

$$x+2-4$$

Subtract 4 from 2 to get -2.

$$x-2$$

Differentiate w.r.t. x

$$1$$

Steps Using Derivative Rule for Sum

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

Subtract 4 from 2 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.

$$1$$