Let's differentiate the function [tex]\( f(x) = (\sqrt{x} - 2)(\sqrt{x} + 2) \)[/tex].
1. Simplifying the Function:
[tex]\[ f(x) = (\sqrt{x} - 2)(\sqrt{x} + 2) \][/tex]
Notice that this is a difference of squares:
[tex]\[ f(x) = (\sqrt{x})^2 - 2^2 \][/tex]
[tex]\[ f(x) = x - 4 \][/tex]
2. Differentiate [tex]\( f(x) \)[/tex]:
Differentiate [tex]\( f(x) = x - 4 \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ f'(x) = \frac{d}{dx}(x - 4) \][/tex]
[tex]\[ f'(x) = 1 \][/tex]
So, the derivative of [tex]\( f(x) \)[/tex] is [tex]\( f'(x) = 1 \)[/tex].
Now, let’s evaluate the given options to see which ones represent the derivative:
Option a:
[tex]\[ \sqrt{x}(\sqrt{x} - 2) + \sqrt{x}(\sqrt{x} + 2) \][/tex]
[tex]\[ = \sqrt{x} \cdot \sqrt{x} - \sqrt{x} \cdot 2 + \sqrt{x} \cdot \sqrt{x} + \sqrt{x} \cdot 2 \][/tex]
[tex]\[ = x - 2\sqrt{x} + x + 2\sqrt{x} \][/tex]
[tex]\[ = 2x \][/tex]
This is not equivalent to the derivative [tex]\( f'(x) = 1 \)[/tex].
Option b:
[tex]\[ \frac{\sqrt{x} - 2}{2\sqrt{x}} + \frac{\sqrt{x} + 2}{2\sqrt{x}} \][/tex]
Combine the fractions:
[tex]\[ = \frac{(\sqrt{x} - 2 + \sqrt{x} + 2)}{2\sqrt{x}} \][/tex]
[tex]\[ = \frac{2\sqrt{x}}{2\sqrt{x}} \][/tex]
[tex]\[ = 1 \][/tex]
This is equivalent to the derivative [tex]\( f'(x) = 1 \)[/tex].
Option c:
[tex]\[ \frac{\sqrt{x} - 2}{\sqrt{x}} + \frac{\sqrt{x} + 2}{\sqrt{x}} \][/tex]
Combine the fractions:
[tex]\[ = \frac{(\sqrt{x} - 2 + \sqrt{x} + 2)}{\sqrt{x}} \][/tex]
[tex]\[ = \frac{2\sqrt{x}}{\sqrt{x}} \][/tex]
[tex]\[ = 2 \][/tex]
This is not equivalent to the derivative [tex]\( f'(x) = 1 \)[/tex].
Option d:
The option says the derivative is 1 directly, which exactly matches our computed [tex]\( f'(x) = 1 \)[/tex].
Thus, the correct options are:
b. [tex]\(\frac{\sqrt{x} - 2}{2 \sqrt{x}} + \frac{\sqrt{x} + 2}{2 \sqrt{x}}\)[/tex]
d. 1
