Solve (2-sqrt{4+x})(2+sqrt{4+x})

$$( 2 - \sqrt { 4 + x } ) ( 2 + \sqrt { 4 + x } )$$

Evaluate

$$-x$$

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}.

$$2^{2}-\left(\sqrt{4+x}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4-\left(\sqrt{4+x}\right)^{2}$$

Calculate \sqrt{4+x} to the power of 2 and get 4+x.

$$4-\left(4+x\right)$$

To find the opposite of 4+x, find the opposite of each term.

$$4-4-x$$

Subtract 4 from 4 to get 0.

$$-x$$

Differentiate w.r.t. x

$$-1$$

Steps Using Definition of a Derivative

Consider \left(2-\sqrt{4+x}\right)\left(2+\sqrt{4+x}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2^{2}-\left(\sqrt{4+x}\right)^{2})$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4-\left(\sqrt{4+x}\right)^{2})$$

Calculate \sqrt{4+x} to the power of 2 and get 4+x.

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

To find the opposite of 4+x, find the opposite of each term.

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

Subtract 4 from 4 to get 0.

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

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.

$$-1$$