Solve (3-sqrt{x+9})(3+sqrt{x+9})

$$( 3 - \sqrt { x + 9 } ) ( 3 + \sqrt { x + 9 } )$$

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}.

$$3^{2}-\left(\sqrt{x+9}\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

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

Calculate \sqrt{x+9} to the power of 2 and get x+9.

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

To find the opposite of x+9, find the opposite of each term.

$$9-x-9$$

Subtract 9 from 9 to get 0.

$$-x$$

Differentiate w.r.t. x

$$-1$$

Steps Using Definition of a Derivative

Consider \left(3-\sqrt{x+9}\right)\left(3+\sqrt{x+9}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3^{2}-\left(\sqrt{x+9}\right)^{2})$$

Calculate 3 to the power of 2 and get 9.

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

Calculate \sqrt{x+9} to the power of 2 and get x+9.

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

To find the opposite of x+9, find the opposite of each term.

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

Subtract 9 from 9 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.

$$-1$$