Solve (3-sqrt{5+x})(3+sqrt{5+x})

$$(3- \sqrt{ 5+x } )(3+ \sqrt{ 5+x } )$$

Evaluate

$$4-x$$

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}.

$$3^{2}-\left(\sqrt{5+x}\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9-\left(\sqrt{5+x}\right)^{2}$$

Calculate \sqrt{5+x} to the power of 2 and get 5+x.

$$9-\left(5+x\right)$$

To find the opposite of 5+x, find the opposite of each term.

$$9-5-x$$

Subtract 5 from 9 to get 4.

$$4-x$$

Differentiate w.r.t. x

$$-1$$

Steps Using Derivative Rule for Sum

Consider \left(3-\sqrt{5+x}\right)\left(3+\sqrt{5+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}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3^{2}-\left(\sqrt{5+x}\right)^{2})$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(9-\left(\sqrt{5+x}\right)^{2})$$

Calculate \sqrt{5+x} to the power of 2 and get 5+x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(9-\left(5+x\right))$$

To find the opposite of 5+x, find the opposite of each term.

$$\frac{\mathrm{d}}{\mathrm{d}x}(9-5-x)$$

Subtract 5 from 9 to get 4.

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

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.

$$-1$$